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CATEGORICAL STRUCTURES AND THEIR APPLICATIONS

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Proceedings of the North-West European Category Seminar

CATEGORICAL STRUCTURES AND THEIR APPLICATIONS Berlin, Germany

28 - 29 March 2003

edited by W.Gahler & G.Preuss Freie Universitat Berlin, Germany

'World Scientific NEW J E R S E Y • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: UK office:

27 Warren Street, Suite 401-402, Hackensack, NJ 07601 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

CATEGORICAL STRUCTURES AND THEIR APPLICATIONS Proceedings of the North-West European Category Seminar Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-053-X

Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore

Dedicated to Horst Herrlich on the occasion of his 65th birthday

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Horst Herrlich at work

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Contents Preface

xiii

Introduction

xv

Opening Ceremony of the North-West European Category Seminar 2003

xvii

On Boole's Booleanness B. Banaschewski

1

Homeomorphically Closed Nearness Spaces

H. L. Bentley and John W. Carlson

13

The Tensor Product of Orthomodular Posets

Reinhard Borger

29

On a Weak Form of the Blumberg Property Wing-Sum Cheung, Yu-Ting Lin, George E. Strecker and Shiojenn Tseng

41

Saturated Collections of Metrics E. Colebunders, R. Lowen and M. Sioen

51

A Uniform Space Proof of a Metrisation Theorem

P. J. Collins

67

Classification of Closure Operators for Categories of Topological Spaces Dikran Dikranjan, Walter Tholen and Stephen Watson . . .

69

Topological Structures in Logics Werner Gahler

99

The Structure of Affine Algebraic Sets Eraldo Giuli

113

A Twisted Triple Category of Track Commutative Cubes K. A. Hardie and K. H. Kamps

IX

121

xx

CONTENTS Productivity of Coreflective Classes in Some Topological Structures Miroslav Husek

143

A Characterization of Co-Retracts of Functional Structures Roland Kaschek

157

Maximal (Sequentially) Compact Topologies Hans-Peter A. Kiinzi and Dominic van der Zypen

173

Supertopologies as Starting Points for Generalized Continuity Structures Dieter Leseberg

189

Introducing Lagois Correspondences Austin Melton

207

On (B,B)-Projectivity Helga Oltmanns, Valdis Laan, Ulrich Knauer and Mati Kilp

219

On Coalgebras which are Algebras Hans-E. Porst and Christian Dzierzon

227

A Hyperspace Completion for Semiuniform Convergence Spaces and Related Hyperspace Structures Gerhard Preuss

237

Convex Effect Algebras and Partially Ordered Positively Convex Modules D. Pumpliin and H. Rohrl

251

Fibrewise Sobriety Giinther Richter and Alexander Vauth

265

Two Applications of Elementary Submodels to Partitions of Topological Spaces J. Schroder and Steve Watson

285

L-Valued Categories: Generalities and Examples Related to Algebra and Topology Alexander P. Sostak

291

CONTENTS

xi

Horst Herrlich's Publications

313

Bernhard Banaschewski: Two Poems

333

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Preface The North-West European Category Seminar 2003 in Berlin was devoted to Horst Herrlich on the occasion of his 65th birthday and his retirement. The organizers of this seminar, Werner Gahler und Gerhard Preufi, have decided to publish a proceedings volume with additional contributions of invited colleagues from everywhere and to dedicate this volume to Horst Herrlich, one of the leading category theorists of the world. During the preparation of this book Bernhard Banaschewski sent two poems for Horst Herrlich to the editors, which will be presented at the end of this edition in order to demonstrate that mathematicians are often humorous people. A high value is set on the applications of category theory according to the varied interests of Horst Herrlich. They include the following areas: 1) Algebra (Dierzon, Kilp, Knauer, Laan, Oltmanns, Porst, Pumplun, Rohrl). 2) Computer Science (Kaschek, Melton). 3) Fuzzy Structures and Logic (Gahler, Sostak). 4) Order Theory (Banaschewski, Borger). 5) Topology: a) Homotopy Theory (Hardie, Kamps). b) Fibrewise Topology (Richter, Vauth). c) Special Topological Constructs (Bentley, Carlson, Cheung, Colebunders, Collins, Dikranjan, Giuli, Husek, Kiinzi, Leseberg, Lin, Lowen, PreuC, Tholen, Schroder, Sioen, Strecker, Tseng, Watson, van der Zypen). We are very grateful to Mrs. Barbara Wengel for collecting the papers and to Mr. Attila Kimpan for preparing them according to the rules of the publisher. Furthermore, we would like to thank Professor Jorg Raasch for taking a picture of Horst Herrlich at work. Last but not least we are indebted to the scientific publisher, Dr. Jitan Lu, from World Scientific for quick publication.

Gerhard Preufi

Werner Gahler

Xlll

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Introduction As is well-known category theory was created by S. Eilenberg and S. Mac Lane in 1945. For a considerable time after this event category theory served mainly as a suitable language for formulations in the framework of homological algebra and algebraic topology. The next decisive notion, namely that of an adjoint functor, was introduced in 1958 by D. M. Kan, though the closely related concept of a universal arrow has been studied by P. Samuel ten years earlier, at least in the framework of Bourbaki's structures. The equivalence of the concepts of adjoint functors and universal arrows (now defined in the realm of category theory) was proved explicitely by J. Sonner in 1964. Also in the sixties of the 20th century the study of applications of adjoint situations were intensified, e.g. reflections and coreflections (P. Freyd, J. Isbell, H. Kennison, H. Herrlich and others) or cartesian closedness (E. Binz, M. Katetov, N. Steenrod and others). In 1971 the field of categorical topology has been established definitively by H. Herrlich and O. Wyler. Its aim is the study of various categories being interesting for topologists, and the relations between them by means of categorical methods. The so-called topological constructs take up a considerable part of this volume. Approximately at the same time theoretical computer scientists became interested in category theory, e.g. in the realm of automata theory (J. A. Goguen, M. Arbib and E. G. Manes, H. Ehrig and M. Pfender). Another discipline in which categorical methods are used is fuzzy theory introduced by A. Zadeh in 1965. Fuzzy theory is related to control theory and to neural networks and provides a lot of contributions to category theory, to different types of logic, and to different types of topological structures (B. Schweizer, A. Sklar, E. Colebunders, R. Lowen, D. Eklund, W. Gahler, U. Hohle, A. P. Sostak and others). It is not the intention of this introduction to give the reader a detailed idea of the historical development of category theory and its applications but much more this book should be a useful guide to learn a lot about the latest development of categorical methods in various fields of its applications.

xv

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Opening Ceremony of the North-West European Category Seminar 2003

Dear colleagues, dear Horst, The organizers of the North-West European Category Seminar 2003, Werner Gahler and I, welcome you very cordially in Berlin. This year, the seminar is devoted to our colleague (and friend of many of us) Horst Herrlich on the occasion of his 65th birthday and his retirement. Let me use this opportunity to give a short survey of his mathematical career. In 1956 (nineteen years after his birth in Berlin) he studied mathematics and biology mainly at the Free University with the goal to become a school teacher. Indeed, he was a school teacher from 1961-1963, but already in 1962 he got his PhD from the Free University in the age of 25 for his thesis "Ordnungsfahigkeit topologischer Raume", and in 1963 he became an assistant in Mathematics at the same university. The students liked him because he was always happy and helped them in their mathematical problems. I know this very well, since I was one of these students, and I am very grateful for his supervision. In 1965 he finished his habilitation thesis on "£-kompakte Raume" and became a "Privatdozent" in 1966 and a professor in 1969. One of his first courses was entitled "Topologische Reflexionen und Coreflexionen" and appeared in 1968 as a book in the Springer Lecture Notes in Mathematics. It is very famous because it is fundamental for the so-called Categorical Topology which started in 1971, mainly by Horst Herrlich and Oswald Wyler. Also at the beginning of his career Horst was invited to be a Visiting Assistant Professor at the University of Florida in Gainesville from 1966 to 1967 and from 1969 to 1970. There he met George Strecker and a life-long friendship started and culminated 1973 in the appearance of the joint book "Category Theory" which is and was a must for every category theorist. This book has been enlarged in 1990 (together with J. Adamek) and appeared under the title "Abstract and Concrete Categories". In 1970 Horst went to Bielefeld and 1971 to Bremen where he retired last year as a full professor in Mathematics. During his time in Berlin 18 papers appeared in well-reputed journals and two papers in Conference Proceedings. This period is also characterized by close connections to the topological school in the Netherlands. I mention here: J. de Groot, J. van der Slot, E. Wattel and last not least George Strecker. xvii

xviii

OPENING CEREMONY

In the seventies Horst published 42 papers, some of them with coworkers. Among them are: J. Adamek, B. Banaschewski, H.L. Bentley, W.W. Comfort, V. Kannan, M. Rajagopalan, W.A. Robertson, G. Salicrup, G. Strecker, and R. Vazquez. In this period he developped not only the fundamentals of Categorical Topology (including topological functors, initial and final completions, and cartesian closedness of topological constructs) but also a new foundation of General Topology by axiomatizing the intuitive concept of nearness between an arbitrary collection of sets. The resulting nearness spaces are essential for the extension theory of topological spaces. His completion of nearness spaces is known today as Herrlich completion. Furthermore, he recognized the importance of factorization structures in Category Theory and developped e.g. dispersed and light factorization structures. In the eighties 37 papers appeared together with a book on topology consisting of three volumes where the last volume is entirely devoted to nearness spaces. His book demonstrates that H2 (= Horst Herrlich) is also an excellent teacher. Since the development of Categorical Topology was very rapid, H2 published a paper entitled "Categorical Topology 19711981" with merely 700 references and several open problems. Futher, he invented Universal Topology - a counterpart of Universal Algebra. For topological constructs, Horst established another convenient property besides cartesian closedness, namely heriditariness, i.e. the existence of one-point extensions. In Category Theory he made precise the difference between algebra and topology. Additionally, he studied Galois connections in a joint paper with M. Husek in 1984, and then in a pure categorical manner in 1990. In the nineties the contacts to friends in Belgium, in particular to R. Lowen and E. Lowen-Colebunders, were intensified, and some joint papers appeared, e.g. on convergence or on improvements of topological constructions. Approximately from the middle of the nineties up to the presence H2 has also worked in the interactions of axiomatic set theory and general topology and got highly interesting results, e.g. on the axiom of choice and the Boolean prime ideal theorem. In the same period several historical overviews on "Categorical Topology", on "Uniform, proximal and nearness concepts", and on several themes in connection with Felix Hausdorff's collected works (together with coworkers) appeared. Summarizing Horst has created an impressive and stimulating work of 165 papers1, six books (mentioned before), and a mathematical biography on G. Salicrup. Furthermore, he has edited 14 conference proceedings (with various coeditors). l

ln the meantime there are 170 papers (cf. p. 313 ff).

OPENING

CEREMONY

xix

It's time for finishing this overview and saying: Hearty congratulations to your birthday, dear Horst! Remain in good health and have good ideas in the future for the benefit of all of us!

Gerhard Preufi

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Upper row (from left to the right): M. HuSek, G. Gutierres, E. Colebunders, R. Lowen, Ch. Dzierzon, H. Herrlich, G. Preuss, H.-E. Porst, J. Schroder, R. Brandt, G. Richter Lower row (from left to the right): H. Kamps, M. Sioen, T. Kubiak, Ch. Schubert, U. Knauer, D. Leseberg, R. Borger, W. Gahler, T.-U. Jandek

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ON BOOLE'S BOOLEANNESS

B. BANASCHEWSKI Department of Mathematics & Statistics McMaster University Hamilton, Ontario, Canada L8S 4K1 E-mail: [email protected] This paper provides an analysis of the formalism presented in Chapters II and III of Boole's "Laws of Thought", showing that it amounts to describing a particular kind of pseudocomplemented meet-semilattice, very far from Boolean algebras. In addition, several identities and identical implications are considered which do turn these semilattices into Boolean algebras. Mathematics Subject Classifications (2000): 06A12,03G99,03G05 Keywords: Boolean algebras,pseudocomplemented meet-semilattices,the laws of excluded middle and double negation as identities

As is well known, George Boole's seminal work in symbolic logic did not quite introduce the concept of Boolean algebra as now understood but rather a more rudimentary system of which it is a streamlined and more evolved version: "originated by Boole, extended by Schroder, and perfected by Whitehead" (Sheffer[7]). This article, going back to the beginnings, presents an analysis of the precise algebraic nature of Boole's own original system as defined by his explicitly given operations and laws. We take this to be presented in its definitive form in Boole[2], given the fact that Boole himself viewed this as the more mature version of its forerunner Boolefl]. Below, all reference to Boole[2] are given by chapter and section numbers. Special note should be taken of the above emphasis on explicit: it is true that, in several instances, Boole allows himself forms of reasoning not covered by his explicit stipulations, and it could perhaps be argued that the latter were not intended as the definition of an algebraic system in our precise sense, so that taking him all too literally might run the risk of misconstruing the text - but an author's intentions are one thing and explicit formulations another, and we believe the latter well worth 1

2

BANASCHEWSKI

analyzing, all other considerations notwithstanding. It turns out, and may come as a bit of a surprise, that Boole's specifications describe algebraic systems that are actually quite far from Boolean algebras proper (Proposition 1) although they do lead to the latter by the addition of the axiom expressing either the Law of Double Negation or the Law of the Excluded Middle (Proposition 2), neither of which, somewhat curiously, appears among the explicitly listed laws. We augment this result by discussing some further conditions equivalent to the Law of the Excluded Middle in this context (Proposition 3). Boole defines what amounts to an algebra of classes by introducing one binary operation, multiplication (xy), two partial binary operations, addition (x+y) and subtraction (x — y), and two distinguished elements, 0 and 1, motivated, respectively, by class intersection, union of disjoint classes, the exclusion of members of one class from a bigger class, the empty class and the universal class. The laws explicitly stated concerning multiplication say, in modern language, that we have an idempotent commutative monoid with zero, 1 being the unit and 0 the zero (116, 119, III 13). It should be explained that, although the familiar expression of associativity,

x(yz) = (xy}z does not actually occur in the text, it may nonetheless be directly inferred from the statement (II7) that zxy and xyz represent the same class since the natural reading of this, involving consistent bracketing,

z(xy) = x(yz)

or

(zx)y = (xy)z ,

together with commutativity yields the desired identity. As to the (really not relevant) inconsistent bracketing,

z(xy) = (xy)z

or

(zx)y - x(zy)

we observe that the first produces nothing new, while the second again leads to associativity. Further, expressing Boole's stipulations for the domains of the two partial operations in algebraic terms, x + y is defined for all x, y such that xy = 0, and

x - y for all x, y such that xy = y, while the laws stated for these are (II11 and II13)

x+y =y+x

(1)

BOOLE'S BOOLEANNESS

z(x + y) = zx + zy,

z(x - y) = zx - zy

if x = y + z then x — z — y .

3

(2)

(3)

What is missing here is a law that governs the distinguished elements with respect to + and —, evidently needed since 10 = 0. Obviously, the most natural way to fill this gap is to pose 1 = 0 + 1,

(4)

immediately yielding

x = 0+x for arbitrary x by (2), which in turn has the two consequences

x - 0 = x,

x —x = 0

(5)

by (3). Note that Boole certainly uses the second identity of (5), but without any formal justification. A comment concerning the operations + and — may be in order here. As to +, Boole makes it unambiguously clear in several places that the motivation is given by the union of disjoint classes, and on that basis it must be viewed as a partial operation in the way described here. On the other hand, at a later stage, he does permit himself to write expressions which no longer satisfy his original restriction on + and provides some discussion as to the meaning of taking this liberty. Similar remarks apply to the operation —. Now, it clearly is an important question how Boole's freewheeling extension of the symbolism, ultimately some kind of passage from a partial to a total algebra, should properly be accounted for, but this is an issue quite different from the concern here. We note that this question is dealt with by Hailperin[4] who places it in the setting of semiprime commutative rings, with unit and torsionfree additive group, where class algebra is recaptured in the form of the corresponding algebra of idempotents. This may be an attractive approach to the problem involved, but in the final analysis it takes it as settled that the notion Boole introduces in Chapters II and III is Boolean algebra proper; our concern, on the other hand, is the prior question to what extent this is actually the case, provided one specifically focuses on what is explicitly stated. Going somewhat beyond Boole himself, it might be worth adding here, again concerning +, that most of Boole's successors (the exception being Venn) decided to deviate from him and to take + as a total operation; indeed, it seems this step was seen as the great advance over Boole's system,

4

BANASCHEWSKI

as indicated by the extensive discussion of this point by Schroder [7], more recently reiterated by Kneale[6]. There is one further detail that requires comment. After introducing his operations and laws, Boole compares his algebra of logic with the usual algebra of numbers (1114,15). Noting that the only numbers for which z2 = x are 0 and 1, he envisages an algebra of number symbols restricted to taking on only these two values and then claims that this will be formally the same as his algebra of logic (II15). This is a dubious statement which we feel free to ignore for our purposes since it can serve neither as an encapsulation of the preceding presentation of operations and laws, nor as a meaningful augmentation of the principles laid down there: for instance, if x and y only range over {0,1} then xy = 0 implies x = 0 o r y = 0 - a n implication Boole surely would not want to claim for his algebra of logic. For similar criticism of II 15 we refer to Kneale[4]. In the following, then, we regard a partial algebra with the above described operations and laws as the embodiment of Boole's explicit formulations in Chapters II and III, and call it a Boolean system. Our first aim is to relate this to more familiar notions. To begin with, note that any Boolean system is a meet-semilattice in which x < y iff x = xy and x Ay = xy, with 0 and 1 as the bottom and top, that is, zero and unit in the partial order sense. In any meet-semilattice with 0, x and y are called disjoint if x A y = 0; further, whenever elements x and y have a join (= least upper bound) this will be denoted xVy&s usual, and it will be called distributive if z A (x V t/) = (z A x) V (z A y) for all z; finally, we say a meet-semilattice with 0 has distributive disjoint joins provided any pair of disjoint elements has a join and this is distributive. Also, recall that a meet-semilattice is called pseudocomplemented if it has a zero 0 and any element x has a pseudocomplement, that is, there is an element x*, necessarily unique, such that x A y = 0 iff y < x*, for all y. Now we have Proposition 1. Any Boolean system B is pseudocomplemented such that x* = 1 — x for any x € B, and for any disjoint x,y 6 B, x V y — x + y. Conversely, any pseudocomplemented meet-semilattice with unit and distributive disjoint joins is a Boolean system with xy = x/\y, x + y = x\/y for any disjoint x and y, and x — y = x A y* for any x > y.

Proof. If xy = 0 in B then y(l — x) = y — yx = y and hence y < I — x-

BOOLE'S BOOLEANNESS

5

conversely, this implies xy = xy(l —x)= y(x — x2) = y(x — x) — 0

since x — x = 0 by (5). This shows x* = I — x. Further, for any disjoint x and y

x(x + y)=x,

y(x + y) = y

so that x, y < x + y; on the other hand, if x, y < z then x = xz and y = yz, and therefore

x + y = xz + yz = (x + y)z , showing x + y < z. It follows that x + y = x V y. For the converse, define xy = x A y for all x, y, x + y = x V y for any disjoint x and y, and x — y = x A y* for any x > y. Then, clearly, (1) and the first identity of (2) hold. Further, for any x > y, z(x — y} = z/\x/\y* while zx — zy = z A x A (z A y)*

and since y* < (z A y)* and z A (z A y)* < y* by the properties of pseudocomplements, these two are equal, which proves the second identity of (2). Next, if x = y + z = y V z for disjoint y and z, then x-z = (yVz)/\z*=yAz*=y, the last step since y A z = 0, showing (3). Finally, (4) is obvious.

D

The second part of Proposition 1 immediately leads to the following Corollary 1. Any pseudocomplemented bounded distributive lattice, and in particular any Heyting algebra, is a Boolean system. In particular, then, any Boolean algebra is a Boolean system, but of course this corollary covers considerably more than that. In addition, it should be noted that a Boolean system need not even be a lattice , as the following example shows: For any meet-semilattice M with zero and unit 1, let B result from M by adding a new zero 0, Then B is pseudocomplemented:

6

BANASCHEWSKI

Further, any disjoint pair x, y e B has trivial join: since x A y = 0 if B iff x = 0 or y = 0 we have 1 = 01 a {y(* x Vy = > ^~ ' \x(y = 0)

and these joins are clearly distributive. Hence, by Proposition 1, B is a Boolean system, and whenever M is not a lattice B is not a lattice, either. Remark 1. It may be worth pointing out that our adoption of the condition 0+1 = 1 is crucial for the first part of Proposition 1: indeed OVl = 1+0 means 1 = 1 + 0. Given that Boole's successors considered the shift from partial to total + as a particularly significant advance, it should be emphasized that this step alone does not create Boolean algebra by any means, as the above corollary makes very clear. Thus, when Schroder[7] presented his "extended" version of Boole's algebra of logic, describing a full-fledged bounded complemented distributive lattice in current parlance, he introduced more modification of the original Boolean formalism, as laid out in II and III, than merely the totality of +. The obvious question exactly what makes a Boolean system actually a Boolean algebra, meaning: i n its standard partial order, is answered by Proposition 2. The following are equivalent for any Boolean system B: (i) B is a Boolean algebra. (ii) B satisfies the Law of Double Negation: 1 — (1 — x) = x, for all x e B. (Hi) B satisfies the Law of the Excluded Middle: x + (1 — x) = I , for all x£B. (iv) B satisfies the Extended Law of the Excluded Middle: x = xy + (x — xy) for all x, y 6 B. (v) B is a lattice such that x V y = x + y(l — x), for all x, y € B. (vi) For all x, y e B, (x - xy) + y = x + (y - xy). Proof, (i) =» (ii). Trivial. (ii) =>• (Hi). In any pseudocomplemented semilattice, (a V &)* = a* A 6*

for any distributive join that may exist, as an immediate consequence of the definition of pseudocomplement. Hence, by Proposition 1, the present

BOOLE'S BOOLEANNESS

7

hypothesis yields x + (1 - x) = x V x* = (x V x*)** = (x* A x**)* = 0* = 1, as desired. (iii) => (iv). Multiply the given condition for y by x. (iv) => (v). If x, y < z so that x = xz and y — yz then

(x + (y - xy))z = xz + (yz - xyz) = x + (y - xy) showing x + (y — xy) < z. On the other hand,

x(x + (y - xy)) = x + x(y - xy) = x by the general rules and

y(x + (y- xy)) = yx + (y - xy) = y by (iv), hence x , y < x + (y — xy) and in all then xVy = x + (y — xy). (v) =>• (vi). Immediate since y V x = x V y. (vi) =$• (v). The given condition implies

(x - xy) +xy = x(x + (y - xy)) = x, hence (iv) and consequently (v) as already shown, (v) =>• (i). We first prove distributivity: z A (x V y) = z(x + (y — xy)) = zx + (zy — zxzy) = zx V zy = (z A x) V (z A y).

Further, x V y = x + (y — xy) implies 1 = x + (1 — x) for y = 1, hence 1 — x is the complement of x, and J5 is a Boolean algebra. D Remark 2. The maps

( x , y ) >->x + y(l -x),

(x, y) t-+ x(l - y) + y

are the two obvious ways of extending + in a Boolean system to a total operation, and (vi) then says these extensions are the same iff the system is a Boolean algebra. Remark 3. The following result places the equivalence of (i) and (ii) into a considerably wider context: For any pseudocomplemented meet-semilattice S with unit, T = {x £ S | x = x**} is a Boolean algebra with 0,A, and 1 as in S, join given by (x* Ay*)*, and complement x* for x.

8

banasckewski BBB kewskiwskisANASCHEWSKI

For the first part, we only need to show that x A y G T whenever x,y € T since 0 = 0** and 1 = 1** trivially. Now x A y < x,y implies (x A y)** < x** = x and the same for y, showing (x A y}** < x A y, the non-trivial part of the identity claimed. Next, since x* A y* < x*,y* we have x, y < (x* A y*)* for any x, y e S, and if x, y < z in T then z* < x* A y* so that (a:* Ay*)* < * * * = * , showing (x* A y*)* = x V y. Further, for any x £ S, (x*)** = x* so that x* € T, and since x V x* = (x* A x**)* = 0* = 1 in T, x* is a complement of x in T. Finally, the resu Iting lattice is distributive:

z A (x V y) A ((2 A x) V (z A y))* = z A (x V y) A (2 A x)* A (z A y)* < (x V y) A x* A j/* = (x V y) A (x V ?/)* =0.

the second step because z A (z A x)* < x* since z A (z A x)* A x = 0. It follows that

z A (x V y) < ((z A x) V (x A y))** = (z A x) V (z A j/), the non-trivial part of distributivity. Now, the condition (ii) in the above proposition amounts to saying that B = {x £ B | x = x**}, making B itself the Boolean algebra considered here, showing (i). It should be added that, for pseudocomplemented distributive lattices with unit, the above result is the classical Glivenko Theorem; it is noteworthy that this still holds under the present considerable weaker hypotheses. Remark 4. Passing from Boolean systems to mere pseudocomplemented meet-semilattices with unit, the preceding discussion shows that, already in this setting, (i) and (ii) of Proposition 2 are equivalent. By way of contrast this is no longer the case for (ii) and (iii): the familiar five-element lattice such that 0 < a, b, b < c, a, c < 1

is a pseudocomplemented meet-semilattice such that x V x* = 1 for all x but of course it is not Boolean. This shows that the "additive structure" of a Boolean system B is essential for the Law of Excluded Middle to make B a Boolean algebra.

BOOLE'S BOOLEANNESS

9

Proposition 2 gives equational conditions which make a Boolean system a Boolean algebra; next, we add to these a number of identical implications which have the same import. Proposition 3. The following are equivalent for any Boolean system B: (i) B is a Boolean algebra. (ii) For all x £ B, I — x = 0 implies I = x. (in) For all x, y € B, x(\ — y) = 0 implies x < y. (iv) For all x,y,z £ B, xy < z implies x < z + (1 - y ) ( l — z). (v) For all x, y, z e B, z < xy and x(y — z) = 0 implies x + (y — z) = (x- z)+y. Proof, (i) => (ii). Immediate consequence of the fact that here 1 = x + (1-ar). (ii) => (i). To begin with, y = 0 whenever y(x + (1 — x ) ) = 0 in any Boolean system: the latter implies 0 = xy(x + (1 — x ) ) = xy

as well as xy + (y - xy) = 0

and hence y = 0. Now, applying this to the identity

we obtain that 1 — (x + (1 — x ) ) = 0, and by the present hypothesis this implies 1 = x + (1 — x), proving (i) by Proposition 2. (i) => (iii). Again, since y + (1 — y) — I here we have

x = xy + x(l -y) so that x = xy, that is: x < y, whenever x(l — y) = 0. (iii) =£> (i). The general identity (1 — (1— x ) ) ( l — x) =0 implies that l - ( l - z ) = (1 - (1 - x))x = x, and (i) follows by Proposition 2. (i) => (iv). In any Boolean algebra, x A y < z implies x < z V (~ y) = z V ((~ y) A (~ z))

for the complement of ~ y of y, and this is exactly (iv). (iv) =>• (i). Applying (iv) to the case x = 1, y = 0, and z arbitrary we obtain 1 = z + (1 — z) and hence (i) by Proposition 2.

10

BANASCHEWSKI

(i) =>• (v). In a Boolean algebra, the stated premiss amounts to z = x/\y, and hence ~ z = (~ x)V(~ y), so that yA(~ z) = yA(~ z) andxA(~ z) = z A ( ~ y) from which it follows that x V ( j / A ( ~ 2)) = xVy = (a;A(~ z)) \Jy. (v) =>• (i). Using the given condition for z = x and y = 1 one obtains x + (1 - x) = I so that (i) by Proposition 2. D Remark 5. Note that, at the level of prepositional logic, the above (iii) represents the principle of proof by contradiction: if p and not-q> is false then p implies q. On the other hand (iv) makes z + (1 — y)(l — z) the relative pseudocomplement y —> z, that is, the element t such that x A y < z iff x < t. Finally, (v) reads as a form of associativity if y — z is also expressed as (—z) + y, as Boole indeed does. Remark 6. Interestingly, by way of contrast, there are no identities, or identical implications, which characterize the Heyting algebras among the Boolean systems. This can be shown be exhibiting a Boolean system which is not a Heyting algebra, but which is the union of an ascending chain of Boolean subsystems that are Heyting algebras. For this, take the lattice obtained by adding the chain {^ | n = 1,2,...} to the top of the fiveelement lattice such that

0 < c< a, b < d and remove d. This provides a Boolean system B, and for each n the elements 0, a, b, c, i , . . . , ^, 1 form a Boolean subsystem such that Bn C Bn+i and B is the union of these Bn. Moreover, each Bn is a distributive lattice and hence a Heyting algebra while B itself is not even a lattice. One might add that the relation of Heyting algebras to Boolean systems in general is somewhat analogous to their relation to pseudocomplemented bounded distributive lattices. There, again, there are no identities or identical implications that single out the Heyting algebras - in obvious contrast to the Boolean algebras given by the identity x\/x* = 1 or, alternatively, x** = x. A final comment, concerning "Laws of Thought" and the Law of the Excluded Middle. In his discussion of the intuitive significance of 1 — x (III 14) Boole leaves no doubt that he accepts this law at an informal level, essentially on the basis of its accepted validity for propositions. On the other hand, in Chapter V, he states without comment that 1, expanded with respect to x, gives x + (1 — x) (V12) while later, arguing in general that the sum of the distinct constituents

BOOLE'S BOOLEANNESS

11

of any Boolean expansion is 1, he calls the initial case of the two constituents x and 1 — x "evident" (V14). Now, if this is interpreted as "evident consequence of the laws explicitly stated in Chapters II and III", our discussion shows this is incorrect: it can only be made evident by referring back to the remarks on the intuitive meaning of 1 - x, but that really amounts to the implicit acceptance of x + (1 — x) — I as a further law. Thus, the decisive feature of Boolean class logic, as understood nowadays, made its original appearance in a surprisingly surreptitious way. This is all the more remarkable in view of the pride of place accorded to the counterpart of the Law of the Excluded Middle, the Law of Contradiction rendered in the form x(l — x) = 0, and the great emphasis given to the fact that it can be derived from the basic rules stated earlier: slightly paraphrasing III 15, x = x2 = 0 + x2 implies 0 = x — x2 = x(l — x). That a calculation within his system should yield this particular result Boole considers especially significant because, as he points out, the law in question is "the most certain of all principles" and "the source of all the other axioms" (Boole refers to Aristotle, Metaphysics III, 3 but the reference actually seems to be Metaphysics F, 1005bl8). Given this emphasis, together with the subsequent discussion which referes to the Law of Contradiction as "dichotomy", it seems plausible that Boole might have considered the Law of the Excluded Middle a consequence of, if not actually synonymous with, the Law of Contradiction - which, expressed in slightly different terms would amount to a confusion between the latter and the Principle of Proof by Contradiction. It certainly seems this was a prevalent notion at one time (see Eisler[3] ), before Brouwer's critique of classical mathematics and the subsequent development of intuitionist logic shed new light on it. The exact origins of this error, which is not explicitly stated by Aristotle (Hitchcock[5]), seem well worth investigating. Acknowledgements. I am indebted to D. L. Hitchcock of the Department of Philosophy, McMaster University, for information concerning Aristotle, to J. S. Kirkaldy of the Institute for Materials Research, McMaster University, for stimulating my interest in the question treated here, to G.H. Moore for alerting me to the work of Hailperin, and to Tatjana Schonwalder of the Fachbereich Philosophie und Okonomie, LudwigMaximilians-Universitat Miinchen, for the reference to Eisler[3]. Also, financial assistance by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

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References [1] G. Boole, The mathematical analysis of logic (1847). Reprinted by Basil Blackwell, 1948. [2] G. Boole, An investigation of the laws of the laws of thought (1854). Reprinted by Dover Publications, Inc, 1951. [3] Eisler, Wortertuch der philosophischen Begriffe. Mittler, Berlin, 1907. [4] T. Hailperin, Boole's Logic and Probability. Studies in Logic and the Foundation of Mathematics, Volume 8, North-Holland, 1976. [5] D.L. Hitchcock, Oral communication, March 2003. [6] W. Kneale, Boole and the algebra of logic. Notes and Records of the Royal Society of London 1 2 (1956), 53-63. [7] E. Schroder, Vorlesungen iiber die Algebra der Logik. Teubner Leipzig 1890. [8] H.M. Sheffer, A set of five independent operations for Boolean algebras. Trans. Amer. Math. Soc. 11 (1913), 481-488.

HOMEOMORPHICALLY CLOSED NEARNESS SPACES H. L. BENTLEY Department of Mathematics University of Toledo Toledo, Ohio, 43606, USA E-mail : [email protected]

JOHN W. CARLSON Professor Emeritus Dean Emeritus University of South Dakota

In Memoriam: John W. Carlson (1940-2003) died after this paper was submitted but before it appeared. The concept of homeomorphically closed nearness spaces is introduced and used to characterize the strict TI extensions of a topological space that have the property that each homeomorphism on the underlying space lifts to a homeomorphism on the extension. Mathematics Subject Classifications (2000): 54E17, 54C20. Keywords: nearness space, extension, homeomorphism

Introduction

It is well-known how the Stone-Cech compactification can be constructed by means of the completion of a certain uniform structure. Similarly, nearness spaces (a generalization of uniform spaces) provide a setting par excellence for constructing strict TI extensions of topological spaces. Just as in the case of uniform spaces, nearness spaces have a completion and this completion is the device by means of which every strict TI extension of a TI space can be reconstructed [He74a], [BH76]. While the completion of a nearness space is pleasant in several ways [BH01], it does fail to be functorial in the non-regular case. Thus, for the extension of maps one needs to impose some extra condition in addition to uniform continuity. Several such extra conditions are presented in [BeOl]. In this paper we are concerned with the extension of a map which is a homeomorphism with respect to the underlying topological structure. We define a nearness space W to be homeomorphically closed provided 13

14

BENTLEY AND CARLSON

that every homeomorphism with respect to the underlying topology of W is uniformly continuous. In contrast to the pathology that arises from having to impose complicated extra conditions (such as was done in [BeOl]), we shall show in this paper that for a concrete homeomorphically closed TI nearness space, every homeomorphism (wrt the underlying topology) possesses a unique extension to the completion. Prom that fact, the result advertised in the abstract is an easy consequence. Nearness spaces and extensions of topological spaces This section contains only previously known results. Our purpose here is to formulate the concepts, notations, and results that we shall need for the presentation of our new results that appear in the following sections. Top denotes the category of topological spaces and continuous maps, Tops denotes the full subcategory of Top whose objects are the symmetric1 topological spaces. Nearness spaces were introduced by Herrlich in 1974 [He74a; He74b; He74c; He88]. These "topological-type" structures can be defined using either one of four concepts: uniform covers, near collections, far collections, or micromeric (or cauchy) collections. For our purposes here, the near collections are usually the most convenient. An explanation of our use of the alphabet may be in order. We usually use the letters X or Y to denote topological spaces and the letters W or V to denote nearness spaces. However, this convention is not rigid. In fact, as will be mentioned below, any symmetric topological space is also a nearness space. 1

Definitions

A nearness space is a set W endowed with a concept of certain collections of subsets of W being near, subject to the following five axioms: (Nl) If a collection A corennes2a near collection B then A is near. (N2) If r\A ^ 0 then A is near. 1

A symmetric topological space is one which satisfies x G c\{y} <=$• y g cl{:r}. These are also known as R 0 -spaces. 2 We say that A corefines B provided each member of A contains some member of B as a subset.

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(N3) 0 is near and {0} is not near. (N4) If A V B is near then A is near or B is near. Here we use the notation

(N5) For each collection A of subsets of W , if clA is near then so is A, where c\A — {x e W\ {{x},A} is near} and dA= {clA\A& A}

We say that A is far if it is not the case that A is near. We say that U is a uniform cover provided that the collection {W\U \U €U} is far. Nearness spaces provide a setting, more general than uniform spaces, where uniform continuity can be defined and studied. 2

Definition

A map f : W —> V between nearness spaces is said to be unifomly continuous provided whenever A is near in W, fA = { jA \ A € A } is near in V. Near denotes the category of nearness spaces and uniformly continuous maps. The basic references on nearness spaces are [He74a; He74b; He74c; He88]. Specific results relating nearness spaces to extensions of topological spaces can be found in [BH76; Be92] and surveys of the extensive literature appear in [He83; BHH98]. For some applications of nearness spaces to special classes of maps on the real numbers, see [BH78]. The closure operator defined in (N5) is a topological closure operator and the topological space so determined is called the underlying topological space of W and is denoted by TW. The underlying topological space of a nearness space is always symmetric, and every symmetric topological space has a compatible nearness structure on it defined by (T)

A is near

iff

Dcl.4^0.

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BENTLEY AND CARLSON

Condition (T) defines a full embedding of Tops into Near, and we assume that this full embedding is an actual inclusion3 of Tops as a full subcategory of Near. Thus a symmetric topological space is a nearness space that satisfies condition (T) and between such nearness spaces, uniform continuity coincides with continuity. We call a nearness space W topological provided it satisfies condition (T). We say that a nearness space W is TI provided its underlying topological space is a TI-space. As mentioned above, a nearness space has a completion. A detailed presentation of the completion appears in [He74a]. For our purpose here, we shall need only the definition together with the characterization of the completion that appears in [Be77] (see also [BH01]). The details follow. 3

Definitions

A uniformly continuous map f : W —» V between nearness spaces is said to be 1. initial provided that whenever A is a collection of subsets of W such that fA is near in V, then A is near in W. 2. strict provided that if B is a collection of subsets of V that is far in V there exists a far collection A in W such that c\v fA corefines B. 3. dense provided fX is a dense subset of V. 4. an extension provided f is initial, dense, and is an injection. It turns out that when X and Y are topological, a map / : X —> Y is strict iff { cly fA \ A C X } is a base for the closed subsets of Y. Thus, the concept of strictness of maps between nearness spaces is a generalization of the usual notion between topological spaces. However, it may be worthwhile to point out that for nearness spaces W and V we have / : W -> V is strict

=>•

/ : TW -> TV is strict

but the converse implication does not hold. For example, let X denote IR2 with its usual topology and let W denote H2 with its usual metric uniformity. Then the identity map / : X —* ~VW is strict but / : X —> W 3

In essence, what we have is an alternative axiomatization of symmetric topological spaces by means of the structure consisting of the collections which are near, i.e., those collections that are required to satisfy the five nearness axioms plus condition (T).

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17

is not. To see this, note that a hyperbola together with one of its asymptotes form a two element collection that is near in W but not in X. 4

Definitions

Let W be a nearness space. 1. A nonempty collection A of subsets of W is said to be a cluster provided it is a maximal near collection. 2. W is said to be complete provided that for each cluster A on W there exists x e W such that A={AcW\x£dA}. We are now in a position to state the characterization of the completion W7* of a nearness space W. W* has as its underlying set of points the set of all clusters on W and has a nearness structure defined by: fi is near in W*

<*=*>

U { flu; | u G fi } is near in W

[He74a]. We have a map e : W —> W* defined by e(x) = {AcW\x£c\A}. The completion W* of a nearness space W is determined up to uniform isomorphism [Be77] by the facts that: 1. W* is a complete TI nearness space. 2. e : W —> W* is a strict, initial, and dense map. Thus, if V is a complete TI nearness space and if u : W —> V is a strict, initial, dense map then there exists a unique uniform isomorphism g : V -» W* such that W

•I \V

>

W*

a

commutes. Par abuse de language, in case W is a TI nearness space, we usually shall assume that the canonical map e : W —> W* is an inclusion. No difficulty

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BENTLEY AND CARLSON

will arise from this convention since when W is TI the target restriction e : W —> e[W] is a uniform isomorphism. For a fuller discussion of the completion of a nearness space, see [BH01]. Our main interest here is in the investigation of strict extensions of topological spaces and the extension of maps in that setting. Nearness spaces are a perfect tool for studying strict extensions as Theorem 7 below shows. 5

Definition

A nearness space is said to be concrete provided that every near collection is a subset of some cluster. 6

Proposition [Be75]

A nearness space W is concrete iff its completion W* is topological. As a preamble to the next theorem, we observe that if W is a concrete TI nearness space, then e : TW —> W* is a strict extension of TW (recall that since W* is topological we have W* = T(W*)). The beautiful theorem of Herrlich that characterizes the strict TI extensions of a topological space is the following one: 7

Theorem [He74a; Theorem 6.3]

If X and Y are TI topological spaces and u : X —» Y is a strict extension then there exists a nearness space W with the same underlying set as X and the same underlying topology as X (i.e., TW = X) such that W is concrete and there exists a unique homeomorphism h : Y —* W* such that the triangle W

Y

h

>

W

commutes. In the above theorem, the nearness structure on W is defined by the condition A is near in W iff fl cly uA ^ 0.

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This nearness structure is said to be the one induced by the extension. Section 11 of [BH76] gives a category theoretic formulation of the relationship between strict extensions of topological spaces and completions of nearness spaces. The two categories involved are Ext and Compl which are defined as follows: Objects of Ext are strict TI extensions u : X —> Y of topological spaces and morphisms are pairs (/, g) of continuous maps such that X -^ Y

f X'

u'

>

Y'

commutes. Objects of the category Compl are pairs (W, W*) where W is a TI nearness space and W* is its completion; morphisms of Compl are uniformly continuous maps / : W* —> V* such that fW c V. The precise statement of the relationship between Ext and Compl is the following theorem. For the definition of equivalence of categories in the sense intended in the following theorem see [AHS90; 3.3]. 8

Theorem [BH76; Theorem 11.6]

The category Ext is equivalent to the category Compl under the correspondence described as follows: C : Ext —> Compl E : Compl -> Ext

where C(u : X —» Y) is the nearness space (and its completion) induced by the extension u : X —> Y and we define E(W, W*) to be the strict extension TW —> T(W*) involving the underlying topological spaces.

Extensions of homeomorphisms We are now ready to present one of our main results which is on the extension of a homeomorphism h : TW —> TV to an isomorphism h : W* —> V*.

20

9

BENTLEY AND CARLSON

Definition

If W and V are nearness spaces then we Jet Ti.(W, V) denote the group of all uniform isomorphisms h : W —> V , where the group operator is, as usual, composition of maps. In case W = V we write H(W) = H(W, W) . If W is a subspace of V we let Hw(V) = {h£ H(V) \ hW = W } . Recall that we are regarding symmetric topological spaces as topological nearness spaces, and for a map h : X —> Y between symmetric topological spaces, continuity coincides with uniform continuity. Thus in this case, H(X, Y) is the group of all homeomorphisms h : X —> Y . Hicks and McKee [HM91] have characterized those nearness spaces W for which each permutation is uniformly continuous and in that case H.(W) is the collection of all permutations of W . In the sequel, when the spaces involved are TI and when it simplifies the exposition, we shall assume that an extension u : X —> Y is an inclusion. Thus we shall say simply "Y is a TI extension of X" . 10

Proposition

Let Y be a strict TI extension of a topological space X and let W be the concrete nearness space induced by this extension. Then

Proof: This result is actually an immediate consequence of Theorem 8 above as follows: A homeomorphism h : Y —> V with hX = X delivers an isomorphism (h\X, h) of Ext and since C : Ext —> Compl

E : Compl —» Ext is an equivalence of categories, (h\X, h} induces an isomorphism h:(W,W*)-+

(W,W)

of Compl. Also, h\W = h\X e H(W). Therefore

{h\X

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21

To show the reverse inclusion, let h : W —> W be a uniform isomorphism. Since the completion W* of W is defined purely in terms of nearness concepts, it is evidently the case that h : W —> W "lifts" to a uniform isomorphism h* : W* —> W* . Therefore we have an isomorphism h* in the category Compl. This isomorphism corresponds to an isomorphism (h, h) : (X,Y) -» (X,Y) in the equivalent category Ext. Therefore h € HX(Y} and since h\X = h the proof is complete. d Theorem 10 shows that the uniform isomorphisms on W are precisely the homeomorphisms on X = TW that can be lifted to a homeomorphism on Y . Thus a homeomorphism on X is a uniform isomorphism on W iff it lifts to a homeomorphism on Y . Theorem 10 establishes a one-to-one correspondence between the groups H(W) and H.x(Y). The following theorem shows that these two groups are isomorphic. 11

Theorem

Let Y be a strict TI extension of X and let W be the concrete nearness space induced by this extension. Then H(W) and 7ix(Y) are isomorphic groups. Proof: This result is an immediate consequence of Theorem 8 above and the results in Section 1 1 of [BH76] . The main source of possible confusion here is one of notation. Thus, for the sake of this proof, let the extension be denoted by the more complete notation e : X —> Y. In Section 11 of [BH76], a functor F : Ext —* Top is denned by putting F(e : X —» Y) = X and > 9) — f where (/, g) is as in the following commutative diagram:

X

-!U Y

X'

-> u'

Y'

(Recall the definition of the category Ext that appears before Theorem 8.) One must exercise some care with the functor F since as is shown in [BH76], it is neither full nor faithful, but fortunately for our purposes

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BENTLEY AND CARLSON

here, it is injective on the preimage of homeomorphisms.4 Thus we can proceed as follows: Let S : U(W) -» UX(Y) be denned by S(g) = f where in the above diagram we take X' = X, Y' — Y, and e' = e. Then S(g) = F ( f , g ) and since F is injective on homeomorphisms, S is also an injection. Proposition 10 says that S is a surjection. Therefore S is a bijection. The fact that F is a functor implies that S is a group homomorphism as follows:

=

F(f1,gl)oF(h,g2) D

In order to characterize the strict extensions X —> Y that have the property that every homeomorphism on X can be lifted to a homeomorphism on Y , the following concept is needed. 12

Definition

A nearness space W is said to be homemorphically closed provided every homeomorphism h : TW —* ~TW is a uniformly continuous map h: W -* W. 13

Theorem

Let W be a, nearness space. Then the following two statements are equivalent: 1. W is homeomorphically closed.

2. Proof: That 2 implies 1 is obvious. Assume W is homeomorphically closed. Note that in all cases we have H(W) C H(~fW). Let h £ H(TW). Then h~l £ W(TW). This means that h~l : ~TW —> 1W is a homeomorphism, and since W is homeomorphically closed, it follows that h~l : W —> W is a uniformly continuous map. Therefore h € 7i(W) . LJ 4

Perhaps it is worthwhile to point out a misprint on line 1 of page 174 of [BH76]: The map / : X -> Y should be rather the map / : X -> X'.

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23

The following theorem is the main result of this paper. It characterizes the strict TI extensions X —> Y such that every homeomorphism on X lifts to a homeomorphism on Y . 14

Theorem

Let Y be a strict TI extension of X and let W be the concrete nearness space induced by this extension. Then every homeomorphism on X lifts to a homeomorphism on Y iff W is homeomorphically closed, Proof:

Suppose that every homeomorphism on X lifts to a homeomorphism on Y. Let h 6 H(X) and let h £ H(Y) denote the lift of h. Then h & Hx(Y) which by virtue of Proposition 10 implies that h = h\X belongs to H(W). Hence ft(TW) = U(W) and by Theorem 13 we have that W is homeomorphically closed. Suppose now that W is homeomorphically closed. Then H(W} — and by Proposition 10,

Hence

H(X) = W(TW) = { h\X Thus every homeomorphism on X lifts to a homeomorphism on Y . 15

d

Corollary

Let W be a concrete TI nearness space. Then every homeomorphism on TW lifts to a homeomorphism on W* iff W is homeomorphically closed. We end this section by definitions and a theorem that show how our results can be applied to a continuous map that is not a necessarily a homeomorphism. 16

Definitions

Let X and X' be topological spaces and let f : X —> X' and g : X —> X' be continuous maps. 1. g is called left equivalent to f if there exists h € H(X]

such that

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BENTLEY AND CARLSON

2. g is called right equivalent to f if there exists k £ Ti.(X') such that 9 = kf. 3. g is called equivalent to f if there exists h € 'H(X) and k 6 H(X') such that g = kfh. Clearly, each of the relations defined above is an equivalence relation on C(X, X'), the set of all continuous maps from X to X'. 17

Theorem

Let X, Y, X', Y' be topological spaces such that Y is a strict TI extension of X and Y' is a strict TI extension of X'. Let W and W respectively be the concrete nearness spaces induced by the corresponding extensions. Let f : X —» X' be a continuous map that lifts to some continuous map Y -> Y'. Let g : X -> X'. Then 1. If g is left equivalent to f and W is homeomorphically closed then g lifts to some continuous map Y —> Y'. 2. If g is right equivalent to f and W is homeomorphically closed then g lifts to some continuous map Y —> Y'. 3. If g is equivalent to f and W and W are both homeomorphically closed then g lifts to some continuous map Y —> Y'. 18

Theorem

Let X be a symmetric topological space. Let £ denote the set of all homeomorphically closed nearness structures on X. Then C with the usual order forms a complete lattice.

Examples

Let F : Tops —* Near be a concrete functor such that for each symmetric topological space X we have X = TFX and for each continuous map / we have Ff and / are the same underlying function. Then the image F[Tops] is a subcategory of Near, all of whose objects are homeomorphically closed. This observation is trivial since a homeomorphism, being continuous, is sent to a uniformly continuous map. In this section we will describe five

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25

different classes of nearness spaces that are homeomorphically closed and are determined by a functor in an analogous manner. As a first example, note that every symmetric topological space is homeomorphically closed since in this case continuity coincides with uniform continuity. The functor in this case is the inclusion. 19

Definition [Ca84a]

The Pervin nearness structure on a symmetric topological space X is the one that is defined by the condition: A collection A is near provided cl A has the finite intersection property. The resulting nearness space is denoted byXp. For TI -spaces X , the corresponding Pervin nearness space Xp is induced by the Wallman compactification of X . 20

Definition [Ca84b]

The Lindelof nearness structure on a symmetric topological space X is the one that is defined by the condition: A collection A is near provided cl A has the countable intersection property. The resulting nearness space is denoted by XL . XL may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is a Lindelof extension that is a subspace of the Wallman compactification of X . In the next two examples, the nearness structure is most directly described using uniform covers (recall the definition in Definition 1). 21

Definition [Ca91a]

The point finite nearness structure on a symmetric topological space X is the one that is defined by the condition: A cover U of X is uniform provided it is refined by a point finite open cover of X . The resulting nearness space is denoted by may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is, under suitable conditions (see [Ca91a; Theorem 3.8]), a metacompact extension that is the smallest metacompact subspace of the Wallman compactification of X that contains X .

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BENTLEY AND CARLSON

22

Definition [Ca94]

The locally finite nearness structure on a symmetric topological space X is the one that is defined by the condition: A cover U of X is uniform provided it is refined by a locally finite open cover of X . The resulting nearness space is denoted by may fail to be concrete, but in those cases where it is concrete (and when X is a TI space), its completion is, under suitable conditions (see [Ca94; Theorem 3.12]), a paracompact extension that is the smallest paracompact subspace of the Wallman compactification of X that contains X . 23

Proposition

Let X be a symmetric topological space. homemorphically closed nearness spaces.

Then X P ,X L ,-^PF,^LF are

REFERENCES [AHS90] J. Adamek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories. John Wiley and Sons, New York (1990). [Ba64] B. Banaschewski, Extensions of topological spaces. Can. Math. Bull. 7 (1964) 1-22. [Be75] H. L. Bentley, Nearness spaces and extensions of topological spaces. In: Studies in Topology, Nick M. Stavrakas and Keith R. Allen, editors. Academic Press, New York (1975) 47-66. [Be77] H. L. Bentley, Normal nearness spaces. Quaestiones Math. 2 (1977) 23-43 and 513. [Be91] H. L. Bentley, Paracompact spaces. Topology and its Appl. 39 (1991) 283 297. [BeOl] H. L. Bentley, Extensions of maps from dense subspaces. In: Categorical Perspectives, Jiirgen Koslowski and Austin Melton, editors. Birkhauser, Boston (2001) 151-176. [BH76] H. L. Bentley and H. Herrlich, Extensions of topological spaces. In: Topology, Proceedings of the Memphis State University Conference, 1975, Stanley P. Franklin and Barbara V. Smith Thomas, editors. Marcel Dekker, New York (1976) 129-184. [BH78] H. L. Bentley and H. Herrlich, The real and the reals. General Topology and Appl. 9 (1978) 221-232. [Be92] H. L. Bentley, Strict extensions of TI spaces. In: Recent Developments in General Topology and its Applications, International

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Conference in Memory of F. Hausdorff (1868-1942), W. Gahler, H. Herrlich, and G. Preuss, editors. Akademie Verlag, Berlin (1992) 3345. [BHH98] H. L. Bentley, H. Herrlich, and M. Husek, The historical development of uniform, proximal, and nearness concepts in topology. A chapter in the book: Handbook of the History of General Topology, Volume II, C. E. Aull and R. Lowen, editors. Kluwer Academic Publishers, Dordrecht, (1998) 577-629. [BH01] H. L. Bentley and H. Herrlich, On the strict completion of a nearness space. In: Proceedings of the 8th Prague Topological Symposium 1996. Electronically published at: http//at.yorku.ca/p/p/a/a/00.htm (1999) 11-21. Also published in hard form in: Quaestiones Mathematicae 24 (2001) 39-50. [Ca84a] J. W. Carlson, Pervin nearness spaces. Topology Proc. 9 (1984) 7-30. [Ca84b] J. W. Carlson, Lindelof nearness spaces. In: Categorical Topology, Proceedings of the International Conference held at the University of Toledo, Toledo, Ohio, USA, August 1-5, 1983, H. L. Bentley, H. Herrlich, M. Rajagopalan, H. Wolff, editors. Helderman Verlag, Berlin (1984) 185-196. [Ca91] J. W. Carlson, Metacompact nearness spaces. Topology Proc. 16 (1991) 17-28. [Ca94] J. W. Carlson, Locally finite nearness spaces. Topology Proc. 19 (1994) 63-77. [He74a] H. Herrlich, A concept of nearness. Gen. Topol. Appl. 4 (1974) 191-212. [He74b] H. Herrlich, On the extendibility of continuous functions. General Topology and Appl. 4 (1974) 213-215. [He74c] H. Herrlich, Topological structures. In: Topological Structures I, Proceedings of a Symposium, organized by the Wiskundig Genoot- schap of the Netherlands on November 7, 1973, in Honour of J. de Groot 1914-1972,, Mathematisch Centrum, Amsterdam (1974) 59-122. [He83] H. Herrlich, Categorical topology 1971-1981. In: General Topology and its Relations to Modern Analysis and Algebra V, Proceedings of the Fifth Prague Topological Symposium 1981, J. Novak, editor. Heldermann Verlag, Berlin (1983) 279-383. [He88] H. Herrlich, Topologie II: Uniforme Raume. Heldermann Verlag, Berlin (1988). [HM91] T. L. Hicks and Rhonda McKee, Near-completely homgeneous spaces. Math. Japonica 36 (1991) 159-164.

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THE TENSOR PRODUCT OF ORTHOMODULAR POSETS

REINHARD BORGER FernUniversitdt Hagen Fachbereich Mathematik Liitzowstr. 125 58084 Hagen E-mail: Reinhard. [email protected] Uni-Hagen. de We show that the tensor product of orthomodular posets always exists if the tacit assumption 0 ^ 1 in the definition of orthomodular posets is dropped. This yields a symmetric monoidal category of orthomodular posets. Mathematics Subject Classifications 81P10

(2000): 06C15, 18A40, 18D10,

Keywords; Orthomodular posets, (bi-)additive map, tensor product, representable functor, symmetric monoidal category

Introduction

A major purpose of orthomodular lattices is the attempt to an algebraic treatment of the mathematical foundations of quantum dynamics. The main example is the orthomodular lattice of orthogonal projections (i.e. idempotent hermitian operators) of a (complex) Hilbert space; this lattice is isomorphic to the lattice of closed vector subspaces of the Hilbert space. But this isomorphism cannot be described purely algebraically; in general, the idempotents of a unital (not necessarily commutative) ring do not form a lattice, as opposed to the commutative case, where the idempotents always form a Boolean algebra. Since in general binary joins and meets do not seem to have a physical interpretation, it looks more appropriate to work with orthomodular posets. Usually, an orthomodular poset is defined as a poset with top element 1 and bottom element 0 and some more structure; most authors do not mention explicitly the hypothesis 0 ^ 1 . With this tacit assumption, Foulis and Bennett [1] conclude that the tensor product of orthomodular posets does not exist in general, because they give an example of an orthomodular 29

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poset X (with 0 ^ 1), for which they obtain 0 = 1 in X ® X. Later several authors tried to overcome this difficulty by relaxing the definition. Foulis and Bennett weakened the notion of orthomodular poset to the notion of orthoalgebra and showed that the tensor product always exists if there exists a unitial biadditive map between the given algebras, where they also assumed 0 ^ I in every orthoalgebra. So, if one admits 0 = 1 for orthoalgebras, the tensor product of orthoalgebras clearly always exists. We shall see below that this also holds for orthomodular posets. In comparison, removal of the condition 0 ^ 1 weakens the definition only very slightly; the only new examples are one-element orthomodular posets. It turns out that this modification yields that the tensor product always exists, and then it becomes a bifunctor, which gives rise to a symmetric monoidal structure an the category of orthomodular posets. If one considers orthomodular posets as logics (cf. [4]), one might want to exclude the case 0 ^ 1 because this means incensistency. But from the mathematical point of view it looks easier and clearer if one allows 0 = 1 because then the tensor product becomes a bifunctor, which even satisfies an associative and a commutative law. If we forbid 0 = 1 , then the tensor product is only partially defined, and the formulation of the above statements requires distinctions of several cases. So we prefer to allow 0 = 1 in the definition and ask whether 0 ^ 1 holds in concrete applications. Unfortunately, our proof uses the Adjoint Functor Theorem and does not give insight into the structure of the tensor product; in particular, it does not answer the question of when the tensor product collapses to one point. These questions remain interesting, and in general the structure of the tensor product seems to be quite complicated. On the other hand, we gain the advantage that the machinery of symmetric monoidal categories is applicable.

1. Orthomodular Posets We consider a poset with a least element 0 and an involutive antitone map X —> X, x i-> x-1 (i.e. or-1-1- = x for all x and yL < XL for x < y). Then obviously, 1 := Ox is the largest element of X. For x, y e X we define X-Ly :<& x < y1; in this case we call orthogonal to y. We call X with this data an orthonormal poset if it satisfies the following three axioms: (OMP1) If zJ-z, then x = 0. (P2) If Z-Ly, then the join (supremum) of x and y exists.

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(OMP3) If x < y, then there exists a z with xLz such that y is the join of x and z. Observe that we have not assumed 0 ^ 1 ; but 0 = 1 implies X = {0}. This is a slight simplification of the usual definitions; we shall see soon that it is equivalent to the usual definition. The map x t—» X1- will be called complementation; this is justified by Proposition 1.1 below. The notation x x is not usual but seems more appropriate and suggestive than the common notation x'. For x±.y, the join of x and y is denoted by x + y. Proposition 1.1. The following statements hold in each orthomodular poset X : (i) 0_l_x, x_LO, x + 0 = 0 + x = x for all x € X. (ii) If x-Ly, then y±x and x + y = y + x. (Hi) Ifx-Ly, xJ-2, andy±z, thenx + y±.z, x±.y + z, andx + (y + z) = (x + y) + z. (iv) If xLy, then 0 is the meet (infimum) of x andy. (v) x.LxL and x + x-1 = 1 for all x € X. (vi) If xJLy and x + y = 1, then y = x^ . (vii) If x
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(see Example 1.1 below). From this it is easy to see that our definition coincides with the usual notations (cf. e.g. [4]). But observe that even in an orthomodular lattice for elements x, y with meet 0 and join 1 it does not follow in general that y = x^, i.e. lattice theoretic complements need not be unique. The guiding example used for the foundation of quantum mechanics is the orthonormal poset of orthogonal projections, i.e. idempotent selfadjoint operators on a complex Hilbert space. More generally, for a (unital and associative, but not necessarily commutative) ring R the idempotent elements (i.e. x with x2 = x) form an orthomodular poset with 0,1 as in the ring R, x < y :•*=> xy = yx = x, x-1 := l—x, xLy O- xy = yx = 0, x + y as in the ring R. If R has an involution (i.e. a map R —> R, x i—> x* with x** = x, (x + y)* = x* + y*, (xy)* = y*x* for all x,y e R), then all idempotents stable under the involution (i.e. x with x2 = x* — x) form an orthomodular poset under the inherited operations. Example 1.1. Now let R be the ring of n x n-Matrices over a field K for some non-negative integer n. It is easy to see that A < B is equivalent to im A C im B and ker B C ker A. In particular A < B implies rk A < rk B, and if A < B (i.e. A < B and A ^ B) we even have rk A < rk B. The only (idempotent) matrix of rank 0 is the zero matrix 0, and the only idempotent matrix of rang n is the unit matrix E. Now let us consider different values of n. If n = 0, the only (idempotent) 0 x 0-Matrix is 0, i.e. our orthomodular poset (even lattice) is {0} with O-1 = 0; the unique orthomodular poset (up to isomorphism) with 0 = 1. If n = 1, we have two idempotents 0,1, and X is the lattice {0,1} with 0 < 1, O"1 = 1, I-1 = 0. For n = 2, we see that 0 is the least element and E is the largest element of X, and we have rk 0 = 0, rk E = 2, Q-1 = E, E^- — 0. Moreover, there are some idempotent matrices or rank 1, which are pairwise incomparable, and the complementation A H-> AL = E — A induces a fixed point free involutive map among them. Since they are pairwise incomparable, any two distinct ones of them have join E and meet 0. This yields that X is a lattice, but lattice-theoretic complements are not unique. For n > 3 we claim that X is no longer a lattice; it suffices to consider the case n = 3. Then put

/ i o o\ , / i i o\ o o o , AI := I o o o , \o oo/' \oooj /io o\ „ (ioo\ BQ := o i o , BI := o i o . Vo o o/ VOID/

AQu :=

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Then A0,Ai,B0,Bi are idempotent matrices with rk A^ = 1, rk Bv = 2, Ap < Bv (n,i> € {0,1}). Now assume that the join C of A0,Ai exists. Then we have A^ < C, (/j, e {0,1}) since A0,Ai are incomparable, C < Bv, C £ Bv (since #!_„ ^ £„), i.e. C < Bv (v & {0,1}). In particular, this implies 1 < rk C < 2, which is impossible because rk C must be an integer. Open Problem Let R be a unital ring with involution. Is the set of all idempotents which are stable under the involution always a lattice in the mentioned order? The answer seems to be no, but it might be difficult to find a counterexample. We shall also need the following construction: For an orthomodular poset X and for a € X, consider X\a := {x € X; x < a}. Then X\a is poset in the induced oder with least element 0 and largest element a, and X\a becomes an orthomodular poet under the new complementation x I-* x-1' := (x + a-1)-1, in particular O x ' = a and a1-' = 0. For all x, y £ X\a, x + y in X\a is as in X\\ = X; for a = 0 we see that X\o = {0} is the trivial (one-element) orthomodular poset. 2. Categorical Properties

We want to define the category OMPos of orthomodular posets. In the previous section we denned its objects; so now we shall pay attention to the morphisms. Let X, Y be orthomodular posets and let / : X —» Y be a map. If / preserves the order and the complementation (i.e. f(x-L) = f ( x ) ± for all x € X), then / preserves orthogonality (i.e. if XQ,XI e X, xo-Lxi, then /(zo)-L/(zi))- Moreover, / also preserves 0 and 1, because OJ_0 in X yields /(0)_L/(0) in Y and thus /(O) = 0 by (OMP1) and therefore /(I) =/(O- 1 -) = QJ-= 1. We call an arbitrary map / : X —> Y additive if /(z 0 + xi) = f ( x 0 ) + f ( x i ) holds for all x0, xi € X with x0Lxi, and we call / unital if /(I) = 1. Easy examples show that preservation of order and complementation in general does not imply additivity. Proposition 2.1. For ortho-normal posets X, Y and a map f : X —> Y the following assertions hold: (i) f is a homomorphisms of orthonormal posets if and only if f is additive and unital. (ii) If f is additive, then f yields a homomorphism f : X —> ^|/(i) by codomain restriction.

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Proof, (i) The "only if" part is clear, and for the "if" part it suffices to show that / preserves oder and complementation. For XQ,XI e X with XQ < xi, (OMP3) for X yields an x? 6 X with x0±X2, XQ + x2 = xi, hence f(xQ) < f ( x 0 ) + f ( x 2 ) = f(x0 + z 2 ) = f ( x i ) . For every x e X we have x + or1 = 1 , hence f ( x ) + /(a;-1) = f(x + or1) = /(I) = 1, hence /(or1) = /(a:)1- by (v) and (vi) of 1.1. (ii): For every x 6 X we have x + x-1 = 1 in X, hence /(x) < /(x) + /(a;-1) = /(x + or1) = /(I) in r, i.e. f ( x ) & Y\fW. Thus / induces a map / : X —> y|/(j), which is unital by definition and additive by additivity of / and therefore a homomorphism by (i). D For an orthomodular poset X and for o € X, the inclusion map X\a <—> X is always additive, but unital only in the trivial case a = 1; so it is a homomorphism only in this case. Now we can define the category OMPos, whose objects are orthomodular posets and whose morphism are homomorphisms of orthomodular posets. Clearly, the one-element orthomodular poset O := {0} is a terminal object, since for ever X the unique (constant) map X —> O is a homomorphism. On the other hand, if / : O —> X is a homomorphism, then 0 = 1 in O implies 0 = /(O) = /(I) = 1 in X and thus 0 < x < 1 = 0 and hence x = 0 for all x e X, i.e. X = O. Thus all homomorphisms with domain O are isomorphisms. The category OMPos also has an initial object, namely D := {0,1} with 0 < 1, Q-1- = 1, I-1 = 0, for every orthomodular poset X the map 0 —» 0, 1 i—> 1 is the unique homomorphisms from D to X. The forgetful functor is representable the representing object is S ':— {Q, 1, s, sx} with 0 < s < 1, 0 < s1 < 1, Ox = 1, l x = 0, s11 = s, where s and sx are incomparable. For an arbitrary orthomodular poset X and for x G H, there is a unique homomorphism 5 —> X which maps s to x, namely the map 0 H-> 0, 1 H-> 1, s H-> x, sx H-> a;1. It is easy to see that up to isomorphism (9, -D, 5 are the three smallest orthomodular poset, i.e. every orthomodular poset with at most four element is isomorphic to O, D, or S. Theorem 2.1. The category OMPos is complete, and the forgetful functor OMPos preserves all existing (possibly large) limits. Proof. Preservation of limits follows from representability. For the existence, it suffices to construct products and equalizers. For a set I and a family (Xi)i€i of orthomodular posets, the product Hie/ -^» *s *^e set-theoretic

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product with componentwise order and complementation. If /, g : X —> Y are homomorphisms of orthomodular posets, the equalizer of / and g is the set-theoretic embedding Z <-> X, where Z := {x e X; f ( x ) = g(x)} with order and complementation inherited from X. Then Z obviously satisfies (OMP1), and (OMP2) follows immediately from additivity of / and g; for (OMP3) use (vii) of 1.1 and observe (x0 + x^)1- € Z for XQ, Xi € Z C X, XQ < Xi.

D

Observe that the product of the empty family is the terminal object O. For the sake of completeness, we like to mention that OMPos is also cocomplete, which will not be used further in this paper. The coproduct of a non-empty family (Xi)i€i of orthomodular posets can be constructed as follows. If 0 ^ 1 holds in all Xi, then first take the set-theoretic union of all Xi with order and complementation on each summand (such that elements of distinct summands are incomparable) and divide out the equivalence relation ~, where for x e Xi, y £ Xj, we have x ~ y if either x — 0 (in X^, y = 0 (in Xj) or x = 1, y = 1 or i = j, x = y. Then there is a unique orthomodular structure on the set-theoretic quotient such that the projection is a homomorphism of orthomodular posets. Then this quotient is the coproduct of (^¥j)j 6 /. If Xi = O holds for some i 6 /, it is easy to see that O is a coproduct of (-Xi)ig/. The coproduct of the empty family is the initial object D. The existence of coequalizers can be shown by Adjoint Functor Theorem techniques, but this does not yield much insight into their structure. 3. The Tensor Product For orthomodular posets X, Y, Z, we call a map b : X x Y —> Z biadditive if it is additive in each component, i.e. for all x e X the map Y —> Z, y i-> b(x,y) is additive and for all y e Y the map X —>• Z, x >-> b(x,y) is also additive. Observe that for each x G X we have b(x, 0)±b(x, 0), because OJ_0 in Y; this implies b(x, 0) = 0 by (OMP1) in Z; similarly we have 6(0, y) = 0 for all y € Y. Moreover, we call the above b unital if 6(1,1) = 1 holds. Note that this means unitality as a map from the categorical product X x Y to Z. On the other hand, biadditivity is quite different from additivity on X x Y. Lemma 3.1. For orthomodular posets X, Y, Z and for a map b : X x Y —> Z the following statements are equivalent: (i): b is biadditive and unital, and for every orthonormal poset W and

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every biadditive unital map c : X x Y —> W there exists a unique homomorphism f : Z —> W of orthomodular posets with fb = c. (ii): b is biadditive, and for every orthonormal poset W and every biadditive map c : X x Y —* W there exists a unique additive map f : Z -* W with fb = c. Proof, (i) => (ii): If c : X x Y —> W is biadditive, then c is isotone in both components, hence c(x,y) < c(l, 1) holds for all re € AT, y e Y. Thus we obtain an map c : X x Y —> Wj c (i i i) by codomain restriction, and c is clearly biadditive and also unital by definition. Now (i) yields a unique homomorphism / : X x Y —» W^i^) with / o b = c. Then it is easy to see that the composite Z —> VF| c (i,i) t-> W is the unique additive map / : Z -> W with fb = c. (ii) => (i): At first we show that b is unital. By codomain restriction we obtain an biadditive map b : X x Y —> Z\b(i,i), and the universal property (i) renders a unique additive map I : Z —> £|&(i,i) with Ib = b. But the inclusion map u : Z|b(i,i) "—> Z is also additive with b — ub = ulb, and the uniqueness part of the universal property of (i) gives that ul is the identity. In particular, u is surjective, hence 1 € Z = Z|b(i,i), i.e. 6(1,1) = 1. Now let W be an orthonormal poset and consider an arbitrary biadditive unital map c : X x Y —> W. Then from (i) we obtain a unique additive map / : Z —> W with fb = c. But then / is unital because /(I) = /6(1,1) = c(l,l) = 1; thus / is a homomorphism by 2.1. Moreover, / is the only homomorphism with f b = c because it is even the only addive map satisfying this equation. D Proposition 3.1. Let X be an orthonormal poset. (i) For the initial orthonormal poset D = {0,1} the map bo '• X x D —-> X with bfj(x, 0) := 0 and &o(x,l) := x for all x e X has the universal properties of 3.1. (ii) For the terminal orthonormal poset O = {0}, the unique (constant) map X x O —> O has the universal properties of 3.1. Proof, (i) It is easy to see that 60 is biadditive and unital. For an orthonormal poset W and a unital biadditive map c, there is at most one homomorphism / : X —» W with fb = c, because each such / must satisfy f ( x ) = fb(x, 1) = c(x, 1) for all x € X. On the other hand, the map / : X —> W, x i—> c(x, 1) is obviously additive and unital and therefore a

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homomorphism, which satisfies fb — c because f b ( x , l ) = f ( x ) = c(x, 1) and f b ( x , 0) = 0 = c(x, 0) for all x € X. (ii) The (constant) map X x O —» O is trivially biadditive. For an orthomodular poset W and a biadditive map c : X x O —> W, the equation 0 = 1 in O implies 0 = c(l, 0) = c(l, 1) = 1 in W and hence W =* O, which immediately proves our claim. D Theorem 3.1. For any orthomodular posets X, Y there exist an orthomodular poset Z and a unital biadditive map b : X x Y —» Z such that for each orthomodular poset W and each unital biadditive map c : X x Y —> W , there is a unique homomorphism f : Z —> W with fb = c. Proof. We use the Adjoint Functor Theorem (cf. [2], [3]). Consider the functor F : OMPos —> Set that assigns to each orthomodular poset W the set of all unital biadditive maps from X x Y to W. Then a routine check shows that F preserves products and equalizers and therefore all small limits. In order to verify the solution set condition, fix an infinite cardinal a such that both X an Y and thus also X x Y are of cardinality < a. Then for each orthomodular poset W and each unital biadditive map c : X x Y —> W the set-theoretic image AQ of c is of cardinality < a. If An C W is defined and of cardinality < a, then Bn := AnU{x-L \ x € An C W} is of cardinality < a + a = a, and An+i ••= {xi + . . . + xr | r 6 Nu{0}, xi,...,xr &Bn, Xi^Xj

for i ^ j}

is of cardinality < £ r€Nu{0} ar = 1 + £ reN a < a • N0 = a. Thus W0 := UneNu{o} ^" = UneN u {o} Bn C W is a subset of cardinality ^ X)neNu{0} a =a-

Moreover, Wo is closed under complementation and orthogonal sums; thus Wo is an orthomodular poset in the structure inherited from W, and the inclusion Wo ^-> W is a homomorphisms. On the other hand, by codomain restriction, c induces a unital biadditive map X x Y —> WQ. This shows that every unital biadditive bilinear map X x Y —> W factors over a unital biadditive map X x Y —> WQ via a homomorphism WQ —> W, such that the cardinality of Wo is < a. But there are only a set of cardinals /3 < a, and for each such /3 and each set WQ of cardinality f3 there are only a set of structures on WQ as an orthomodular poset and of maps X x Y —> WQ. Thus up to isomorphism in the comma category there is only a set of unital biadditive maps X x Y —> WQ with WQ of cardinality < a. If 7 is the

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cardinality of this set, the for each set I of cardinality there is a solution set of cardinality < 7*. In particular, F satisfies the solution set condition. Hence F has a left adjoint and is therefore representable. If we choose Z to be the representing object and b : X x Y —> Z to be the element of FZ corresponding to the identity of Z under the natural isomorphism, we see that Z and b have the desired properties. D Observe that the above proof does not give much insight into the concrete structure. In particular, it is not clear whether there exists an n e N such that An = An+i always holds in the proof of the solution set condition. This should give some indications about how complicated it is to describe the orthonormal poset generated by some given subset. By the universal property, in Theorem 3.1 Z and b are unique up to a canonical isomorphism; so we can introduce some notations. For X, Y as in 3.1, we call some chosen Z with the properties there the tensor product of X and y, we write it as X®Y, and we denote the map b : X x Y —> Z = X®Y by (x, y) H-> x <8> y. It follows from standard arguments that ® behaves like a functor in each component, und thus we obtain a functor OMPos x OMPos -» OMPos, (X, Y)^X®Y. From 3.1 we obtain canonical isomorphisms X ® D = X and X O = O for all orthomodular posets X. Lemma 3.2. (i) For all orthomodular posets X, Y there is a unique isomorphism X <8> Y —> Y X with x y >-> y <8> x for all x e X, 7 / 6 Y. (ii) For all orthonormal posets X, Y, Z there is a unique isomorphism (X Y) ® Z -> X ® (Y ® Z) with (x ® y} ® z H-» x ® (y ® z) for all

x ex, y e Y, z e z. Proof, (i) Since the map X x Y —> Y <8> y, (x, y) H-> y ® x is biadditive and unital, the universal property of X ® Y renders a unique homomorphism s : X®Y -» y ®X with s(x ® y) = y <S> x for all x e X, ye Y. Since this argument is symmetric, there is also unique homomorphism t : Y ® X <-* X ® y with t(y <8> x) = x ® y for all x e X, y e Y. Then we have st(y ® x) = s(x ® y) = y ® x and ts(x ® y} = t(y ® x) = x ® y for all x e X, y € Y. Now the uniqueness part in the universal property of the tensor product shows that st :Y <8> X -» Y X and £ s : X < S > y — > X ® y are the identity maps, i.e. s and t are isomorphisms with s"1 = t.

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(ii) : For fixed z e Z, the map XxY -> X(g)(y®Z), (x,y) i-> z(yz) is biadditive; thus there exists a unique additive map hz : X <S> Y —> X ® (y <8> Z) with /i z (x ® y) = a; (y <8> 2) for all x e X, y e y. For z 0 , zi e Z with zo-Lzi it follows that y®zo-Lyzi and a;<8>(y<8>zo)-La;<8>(y<8>zi) for all a; € X, y e y, and for all x e X, y 6 y we obtain hZQ(x®y) + hZl(x z ® (y ® z0) + z <8> (y <8> Zi) = X (y z0 + y <8> zx) = z ® (y (z0 + hzo+zi(x®y). Themap/i 2o +/i Zl : X®Y -» X<8>(y<8>Z), u i-> hzo(u)+hzi(u) is clearly additive; thus the map (X(g>y) x Z —> X(y®Z), (u, z) i-> ftz(u) is biadditive, and the universal property renders a unique additive map s : (X®Y)®Z -> X<8>(yig>Z) with s(u®z) = /i z (u) for all u G Jf<8>y, z e Z. In particular, we have

s((x ® y) (y ® z) for all x £ X, y € y, z e Z, and by tracing the construction backwards we see that s is the only additive map with this property. Moreover we have

s((l ® 1) ® 1) = 1 ® (1 ® 1) = 1 ® 1 = 1, i.e. 5 is unital and therefore a homomorphism. In the same way we obtain a unique homomorphism t : X ®(Y ® Z) —> (X y) Z with t(x ® (y ® z)) = (x ® y) ® z for all x e X, y 6 Y, z £ Z. Going through the construction, the uniqueness part of the universal properties shows that st : X ® (y Z) -» X ® (y ® Z) and is : (X ® Y) ® Z -> ( X y) (g) Z

are identity homomorphisms, therefore s is an isomorphism and t = s 1 .D Theorem 3.2. OMPos is a symmetric monoidal category with tensor product <8> anrf unit object D = {0,1} in the canonical way. Proof. For orthomodular posets X, Y, Z let the associativity map (X ® y) <8> Z —> X (y ® Z) be the unique homomorphism with (x <8> y] ® z i—» x ® (y ® z) for all x 6 X, y € y, z e Z. Obviously, this isomorphism is natural in all arguments. Similarly, the symmetry X ®Y —> Y d§ X is the unique map with x y >—> y ® x for all x 6 X, y e y. Moreover, the isomorphisms X £) —» X are given b y x < E > l | - ^ a ; , x ® 0 i—> 0 for all x e X; its inverse is given by x \—> a; (gi 1 for all x e X. Similarly we have an isomorphism D ® X = X. All these maps are natural in all arguments. The proof of the coherence conditions is technical but straightforward. D

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It is quite easy to see that OMPos is not closed in the structure; the functor OMPos —* OMPos, X H-> X Y has a right adjoint only if Y = D. Indeed, it this functor has a right adjoint, then it preserves the initial object D, i.e. Y = D ® Y = D. References [1] Foulis, D. and Bennett, M.K, Tensor products of orthoalgebras, Order 10 (1993), 271-282. [2] Herrlich, H. and Strecker, G.E., Category theory, Allyn and Bacon, Boston 1973. [3] Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics, Springer, Berlin 1972.. [4] Ptak, P. and Pulmanova, S., Orthomodular Structures as Quantum Logics, Fundamental Theories of Physics 44, Kluwer 1991.

ON A WEAK FORM OF THE BLUMBERG PROPERTY

WING-SUM CHEUNG Department of Mathematics University of Hong Kong Pokfulam Road, Hong Kong email: [email protected] YU-TING LIN AND SHIOJENN TSENG* Department of Mathematics Tamkang University Tamsui, Taipei, Taiwan, 25137, R.O.C. email: [email protected] GEORGE E. STRECKER Department of Mathematics Kansas State University Manhattan, KS 66506 USA email: streckerQmath. ksu. edu

In this paper we modify and extend the techniques used by Blumberg in his classic paper of 1922. In particular, we formulate a weakening of the Blumberg property and show that a wide class of not necessarily metrizable spaces has this property. Mathematics Subject Classifications 54C30, 54C50, 54E52

(2000): 26A15, 26A21, 54C08,

Keywords: real-valued function, (weak) Blumberg property, (weak) Blumberg space, separable T\ Baire space, dense approach, exhaustible approach

"This work is partially financially supported by the National Science Council of the Republic of China under the projects NSC87-2115-M032-004 and NSC88-2115-M032010 41

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CHEUNG ET AL.

1. Introduction In 1922 Blumberg [1] showed that every separable completely metrizable space X has the following surprising property: B: for each real-valued function f : X —» R there is a dense subset D of X such that the restriction f \D is continuous. Since then B has been called the Blumberg property and spaces having it have been called Blumberg spaces. The theory of Blumberg spaces is well-developed. For example, in 1960 Bradford and Goffman [2] showed that every Blumberg space is a Baire space and that the converse is true if the Baire space is metrizable. In 1977 Weiss [5] showed that there are compact Hausdorff spaces that are not Blumberg. In this paper we consider spaces that need not be metrizable and develop a theory for them analogous to that developed in [1]. We define weak Blumberg spaces, show that every separable T\ Baire space is weak Blumberg, and provide examples of non-metrizable weak Blumberg spaces. 2. The Weak Blumberg Property Throughout let (X, r) be a topological space (often denoted by X alone), let R be the real numbers with the usual topology, let Q be the rational numbers and N be the natural numbers. For each p e X, the set of open neighborhoods of p will be denoted by A/"p. Definition 2.1. Let 9^ be a binary relation between open sets and elements, i.e., 91 C T x X. For U a member of T and p an element of X, Uytp means that the open set U has the relation SH to the element p, i.e., (U,p) e IH. We say that the relation 9f is closed if for every subset A of X and every U G T, t/fHs for all s e A imply U9lp for all p e A. Definition 2.2. A partial neighborhood of p, often denoted by N<, is an open set with p in its closure. Lemma 2.1. // a subset T of X has the property that each p & T has a partial neighborhood V< such that V< C\T = 0, then T is nowhere dense in X. Proof. Let V = U{^< : p 6 T}. Then for each p e T we have p e Vf C V which implies that T CV. But Tr\V = 0, soT CV\V, which is nowhere d dense.

BLUMBERG PROPERTY

43

Lemma 2.2. For each f : X —> R and eac/i pair of real numbers r\ and r% with ri < TI the relation 9^ir2 C. T x X given by: p if and only ifpeU and there exists some q £ U such that ri < is a closed relation. Proof. Suppose that A C X and U e T such that E7Sft£jT.2s for all s & A. Then for each such s we have s e t / and there is some q 6 (7 such that ri ^ /( R, i/ien /or even/ pazr o/ reo/ numbers r\,rz, the set of elements p of X that satisfy both (a) Ny\.^ir.2p for every open neighborhood N of p and (b) N K is said to be densely approached at a point p of X if and only if for each e > 0 there exists an open neighborhood Ne of p such that the set of all elements q of Ne for which |/(
44

CHEUNG ET AL.

(b) A subset of a topological space is called residual if and only if its complement is of the first category. Theorem 2.1. For each topological space X a subset R of X is precisely the set of all elements of X that are densely approached by some real-valued function f : X —> R if and only if R is a residual subset of X. Proof. Let / : X —> R. To show that R = {p e X : f is densely approached at p} is residual, we will show that the complement X \ R is a set of the first category. If p e X \ R then there is an e > 0 such that for each open neighborhood N o f p the set {q 6 N : \f(q) — f(p)\ < f} is not dense in N. Thus for each such neighborhood N of p we can choose an open set UN such that UN n N ^ 0 and |/( e for all qeU. Let n , r 2 £ Q such that f(p) - e < ri < /(p) < r2 < /(p) + e. Then C/9l£r2p is false and M9t£ir.2p for each M 6 JVP. Thus p e T/ir2 (as defined in Lemma 2.3), so that X \ R C T-^ (as defined in 'Corollary 2.1) which is a set of the first category. Hence R is residual. To see the converse, if R is residual, then X \ R is of the first category. So there is a countable family Q — {Gn : n 6 N} of pairwise disjoint nowhere dense sets such that X \ R = U Q. Define / : X — > R by

2 Then / is densely approached at p if and only if p € .R.

D

Definition 2.4. A function / : X —» R is said to be exhaustibly approached at a point p of X if and only if an open neighborhood N o f p and an e > 0 exist such that the set of elements q of N where \f(q) — f(p)\ < £ is of the first category. If / is not exhaustibly approached at p then we say that / is inexhaustibly approached at p. Theorem 2.2. For every separable space X and every real-valued function f : X —> R the set consisting of those elements x of X at which f is exhaustibly approached is of the first category. Proof. For each r e Q and n e N let Srn = (r - ±,r + £). If / is exhaustibly approached at q, then there exist a positive number e and an open neighborhood Nq of q, such that f " l [ ( f ( q ) — e, f(q) + e)} n Nq

BLUMBERG PROPERTY

45

is of the first category. Let's pick the set Srn such that f(q) e Srn Q (/(
1

[5rn] : a neighborhood ./Vx of x exists such that / M^Wi] n Nx is of the first category}

Then E = \J{Ern : r £ Q, n € N} is the set of all elements at which / is exhaustibly approached. Now, it is sufficient to prove that every Ern is of the first category since E is a countable union of such sets. Fix Ern. X is separable, i.e., X has a countable dense subset, say P = {j)i : ? e N}. For each x in Ern pick and fix an open neighborhood Nx of x such that f ~ l [ S r n ] n Nx is of the first category. P is dense. So, for each such Nx, we can pick and fix an element p, of P such that pi belongs to Nx. Let £t = {Nx : pi e A^}- Take a member Mj from each nonempty family &. If £» = 0, let Mj = 0. Let M = \J{Mi : i 6 N}. we claim that Ern\M is nowhere dense. Suppose that the claim is not true. Then Ern\M contains a nonempty open subset U. Pick an element y from the set U fl (Ern\M). Then Ny n U is an open neighborhood of y. This implies that there exists an element pk of P such that Pk £ NyC\ U. Thus Pfc £ Mfc 6 £fc, so that pfc belongs to M. Thus, p^ belongs to U PI M. Since M is open, X\M is closed. U C E rn \M C X\M = X\M. Thus, U n M is empty, which contradicts that p/c belongs to U n M. Hence, the claim is true, so that (Ern\M) U {/^[S™] n M; : i e N} is of the first category. Also Ern = (Ern D M) U (Ern\M)

!5™! n ^ : i € N} U (£ rn \M). Thus -Ern is of the first category.

D

If M is a subset of X, we shall use, in connection with approach, the expression "via M" to indicate that the range of p is restricted to M. Definition 2.5. For any set M C X a function / : X —» R is said to be exhaustibly approached at p via M if and only if an open neighborhood N of p and a positive number e exist such that the set of elements q of NnM where \f(q) - /(p)| < e is of the first category; otherwise, / is called inexhaustibly approached at p via M. We know that if A is of the first category, then every subset of A is also of the first category. Thus, if / is exhaustibly approached at p, then / is

46

CHEUNG ET AL.

exhaustibly approached at p via M for each subset M of X. On the other hand, for any M C X, if / is inexhaustibly approached at p via M, then / is inexhaustibly approached at p. Definition 2.6. A function / is said to be densely approached at p via M if and only if for each e > Q there exists an open neighborhood Ne of p such that the elements q of M n Ne for which |/(p) — f(q)\ < e form a dense subset of M n Nf. Lemma 2.4. If M is a subset of X such that p is a limit point of M, the following statements are equivalent. (1) f : X —> R is densely approached at p via M. (2) For every partial neighborhood N< of p such that N< n M has p as a limit point, f [ N < H M] has f(p) as a limit point. Proof. Assume that (1) holds, but (2) does not. Then there is some partial neighborhood N< of p having p as a limit point, but for which f(p) is not a limit point of /[/V< n M]. Then there is some e > 0 and U € Mp such that U H 7V< n M ^ 0 and |/(p) - f(q)\ > € for all q 6 U n N< n M with q =£ p. By (1) there is some Nf e Mp such that {q* £ M n Ne : \f(q*) - f(p)\ < e} is dense in M n Ne. Now W = (U n M n /V<) n (M n 7Ve) is nonempty and open in M n Ne. Also |/(r) — /(p)| > e for all r e W with r ^ p, which contradicts (1). Thus (1)=K2). Now suppose that (1) does not hold; i.e., there is some e > 0 such that for each N 6 A/"p we can pick an open C/jv such that C/w n M n TV ^ 0 and |/(p) - /(»| > e for all 2 € t/jv n M n TV. Let C/ = \J{UN : N e A/"p}. Then C7 is a partial neighborhood of p and |/(p) — f ( x ) | > e for each x e U n M. Thus /(p) is not a limit point of /[[/TlM]. Since p is a limit point of UnM, we have that (2) does not hold. Thus (2)=»(1). D Theorem 2.3. For ever?/ real-valued function f defined on a separable TI space X, there exists a residual subset S of X such that if p is an element of S then the function f is both inexhaustibly and densely approached at p via S. Proof. Let E\ be the set of elements at which / is exhaustibly approached, and let 5i = X\E\. By Theorem 2.2, we know that EI is of the first category. Then S\ is residual and / is inexhaustibly approached at the elements of Si. That is, if q is an element of Si, then for each e > 0 and for each open neighborhood TV of q, there exists a subset MofN such that M is

BLUMBERG PROPERTY

47

of the second category and \f(x} — f(q)\ < f. for all x in M. MnSi = M\E\ is of the second category since M n E\ is of the first category and M is of the second category. Thus, / is inexhaustibly approached at the elements of Si via Si . Case 1. Suppose that / is bounded. Then there exists a real number k such that k > sup f[X}. Now define g : X -> R by: if,

' Then, by Theorem 2.1 the elements of X at which g is densely approached constitute a residual set, say, Sg. For every element r in EI and for every s in Si, fif(r) — g(s) > k — sup f[X] > 0. Each of g and / is densely approached at the elements of Si n Sg via Si. Furthermore, if K = X — Sg, then Sg n Si — X\(K U EI) is a residual set since K and EI are each of the first category. It follows that the elements of Si at which / is densely approached via Si constitute a residual set, say, S. Let EI = S\\S. Then X = EI U EZ U S. This implies that E^ is of the first category. For every element p of S, f is inexhaustibly approached at p via S since E% is of the first category and / is inexhaustibly approached at the elements of Si via Si . Also, / is densely approached at each such p via Si . For every partial neighborhood N< of p, if p is a limit point of N< n S then p is a limit point of 7V< Pi Si. By Lemma 2.4 this implies that /(p) is a limit point of f [ N < PlSi]. That is, every open neighborhood of f(p) intersects f [ N < flSi] in some element other than f(p) itself. For each e > 0 and for each open neighborhood N of p, there exists an element which is distinct from p, say, p* such that p* belongs to N n (N< n Si) and |/(p*) - /(p)| < e/2. It follows that / is inexhaustibly approached at each such p* via S since p* belongs to Si and Si\S = E^ is of the first category. X is a Ti space so {p} is closed. It follows that (N n A/<)\{p} is an open neighborhood of p*. Thus, a subset M of (N n N< n S)\{p} exists such that M is of the second category and for every element q* of M, \f(q*) — /(p*)| < e/2. By the triangle inequality, we have that )/( R by f(x) = f ( x ) / ( l + |/(x)|). Then / is bounded and the properties of dense approach, exhaustible approach and inexhaustible approach of / are preserved by /. So we can proceed as in Case 1. D

48

CHEUNG ET AL.

Corollary 2.2. For every real-valued function f defined on a separable 7\ Baire space X, there exists a dense subset S of X such that if p is an element of S and N< is a partial neighborhood of p then / is inexhaustibly approached at p via 7V< n 5. Proof. By Theorem 2.3, there exists a residual set S of X such that if p is an element of S then / is densely approached at p via S. Thus by Lemma 2.4 if M< is a partial neighborhood of p such that p is a limit point of M< n S, then f(p) is a limit point of /[M< n S]. Given a partial neighborhood 7V< of p, N<\{p} is open since X is T\. Since X is a Baire space S must be dense. So, p is a limit point of N< n S. It follows that f(p) is a limit point of f[N< n S]. So, for each open neighborhood N of p and for each e > 0, there exists an element p* which is distinct from p such that p* belongs to TV n N< n S and |/(p*) - f(p)\ < e/2. Now p* belongs to S, so that by Theorem 2.3, / is inexhaustibly approached at p* via S. Since N< n TV is an open neighborhood of p, there exists a subset M of N< C\Nr\S such that M is of the second category and |/(p*) —f(q)\ < e/2 for all q in M. Thus \f(q) — f(p)\ < f- for all q in M, so / is inexhaustibly approached at p via 7V< n S. D Definition 2.7. We say that X is a weak Blumberg space or that X has the weak Blumberg property if and only if for each real-valued function / denned on X there exists a dense subset D of X such that / is densely approached at p via D for each p € D. Theorem 2.4. The family of weak Blumberg spaces contains both the family of Blumberg spaces and the family of separable T\ Baire spaces. Proof. The first containment is clear from the fact that every subset of a space is dense in itself. Also note that every residual subset of a Baire space is dense. Hence the second containment follows immediately from Theorem 2.3. D Example 2.1. [0,1]2 ° is a non-metrizable weak Blumberg space. Proof. [0,1]2 ° is a separable non-metrizable compact Hausdorff space, and every such space is T\ and Baire. By Theorem 2.4 it is a weak Blumberg space. D Example 2.2. The set of real numbers with the cofinite topology, K c /, is a non-metrizable weak Blumberg space that is not a Blumberg space.

BLUMBERG PROPERTY

49

Proof. Rc/ is clearly T\ and non-Hausdorff, thus non-metrizable. Since a subset of Ec/ is dense if and only if it is infinite, it follows easily that Ec/ is separable and Baire. By Theorem 2.4 it has the weak Blumberg property. Consider the identity function / : Rc/ —> E. If f\D were continuous for any dense subset D of E c /, then D would be an infinite Hausdorff subspace of E c /, which is impossible. Thus Ec/ does not have the Blumberg propertp

References [1] Henry Blumberg, New properties of all real functions. Trans. Amer. Math. Soc. 24 (1922), 113-128. [2] J. C. Bradford and Casper Goffman, Metric spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc. 11 (1960), 667-670. [3] Murray Eisenberg, Topology. Holt, Rinehart and Winston, New York, 1974. [4] James R. Munkres, Topology: a first course. Prentice-Hall, Englewood Cliffs, N. J. , 1974. [5] William A. R. Weiss, The Blumberg Problem. Trans. Amer. Math. Soc. 230 (1977), 71-85. [6] H. E. White, Jr., Topological spaces in which Blumberg's theorem holds. Proc. Amer. Math. Soc. 44 (1974), 454-462. [7] Albert Wilansky, Topology for analysis. Xerox College Publishing, Waltham, Mass., 1973.

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SATURATED COLLECTIONS OF METRICS

E. COLEBUNDERS1 AND M. SIOEN2 Vrije Universiteit Brussel Pleinlaan 2 1050 Brussel, Belgie E-mail1: [email protected] E-maii : [email protected]

R. LOWEN Universiteit Antwerpen Middelheimlaan 1 2020 Antwerpen E-mail: [email protected]

We show that a whole bunch of well known topological constructs have isomorphic descriptions by means of sets structured by some collection of (generalized) metrics of a certain type together with suitable morphisms. The classification of the categories listed in the table at the end of the paper, is based on different saturation conditions acting on certain collections of generalized metrics. These conditions are composed out of different building blocks. For instance on collections of quasi-pre-metrics we encounter saturation conditions that are combinations of the following properties: prime ideal, ideal, locally qualified saturation, uniform qualified saturation, locally quantified saturation and uniform quantified saturation. Combinations of these enable us to characterize the constructs in the first colum of the summarizing table at the end of the paper: pretopological spaces, preclosure spaces, semi-quasi uniform spaces, pre-approach spaces, quasi -pre metric spaces. The more important subconstructs of these examples, such as topological spaces, completely regular spaces, uniform spaces, proximity spaces, approach spaces and metric spaces, are characterized using essentially the same building blocks of saturation properties, restricted to particular subclasses of generalised metrics. Mathematics Subject Classifications (2000): 54B30, 18B99, 18E20 Keywords: generalized metrics, saturation, topology, pretopology, closure, uniformity, proximity, approach space

51

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COLEBUNDERS ET AL.

1. Introduction

Although metrics lie at the basis of many topological theories, this knowledge is mainly of a conceptual nature and often rather vague. A notable exception is Unif, which can be represented as a category, the objects of which are endowed with a so-called uniform gauge of metrics [15]. In the same way Ap is known to be characterized as a category, the objects of which are endowed with a gauge of quasi-metrics [19]. In both cases the gauges are ideals satisfying some further saturation conditions. The differences between the two examples lie not only in the fact that in the first case metrics are used and in the second case quasi-metrics. The saturation conditions for the gauges have an essentially different component: for uniform spaces, what we will call, a uniform qualitative saturation is needed and for approach spaces a local quantitative saturation is involved. In this paper we show that a whole bunch of well known topological constructs have isomorphic descriptions by means of sets structured by some collection of (generalized) metrics of a certain type together with suitable morphisms. In order to describe the categories listed in the table at the end of the paper we will need several different saturation conditions acting on certain collections of generalized metrics. These conditions will be composed out of different building blocks. For instance on collections of quasipre-metrics we will encounter saturation conditions that are combinations of the following properties that will be explicitely described in section 3: prime ideal, ideal, locally qualified saturation, uniform qualified saturation, locally quantified saturation and uniform quantified saturation. Combinations of these will enable us to characterize the constructs in the first colum of the summarizing table at the end of the paper: pretopological spaces, preclosure spaces, semi-quasi uniform spaces, pre-approach spaces, quasi -pre metric spaces. The more important subconstructs of these examples, such as topological spaces, completely regular spaces, uniform spaces, proximity spaces, approach spaces and metric spaces, will be characterized using essentially the same building blocks of saturation properties, restricted to particular subclasses of generalised metrics. 2. Preliminaries

Given a function d : X x X —» [0, oo], we will say that d is a quasi-pre-metric if it is zero on the diagonal, we will drop "pre" if d satisfies the triangle inequality and we will drop "quasi" if d is symmetric. Note in particular that we do not suppose a metric to be finite, nor do we suppose it to have

METRICS

53

a Hausdorff underlying topology, but we do suppose it to be symmetric. If x £ X and r € [0, oo] then we put Bd(x, r) := {y e X \ d(x, y) < r}. We consider the following constructs of (quasi-)(pre-)metric spaces: Met9p, Metp, Met9, Met and Met*6 respectively the construct of quasipre-metric spaces, of pre-metric spaces, of quasi-metric spaces, of metric spaces and of totally bounded metric spaces. We use Met* as a generic notation for any of these, where * stands for any of the possible superscripts, including the empty superscript. In all cases the morphisms are the non-expansive maps. As usual, on any given set X, Met*(X) stands for the partially ordered set of all ^-metrics on X where the partial order simply is the pointwise order. By a downset in Met*(X) we mean a non-empty subset S C M.et*(X) such that d£S,e& Met*pQ, e < d => e e S, i.e. simply a downset in the partially ordered set Met* (A). As a blanket reference for categorical notions, we refer to [1, 17, 24]. Concerning the order in the fibres of a topological construct, we follow the convention taken in [24] of considering the finer of two objects as the larger one. 3. Saturation conditions We aim to prove that all categories described in the table at the end of the paper have objects, the structure of which can be described by means of downsets in Met* (A), for some of the * denned earlier, case by case fulfilling supplementary conditions, and that remarkably in all cases the morphisms are described in a similar way. We now introduce a number of saturation properties for downsets T> of Met*(X). (IDL) T> is an ideal in Met*(X) if d, e e J =$> d V e e J. (PIDL) T> is a principal ideal in Met* (X) if it is an ideal generated by a unique element, i.e. an ideal of the type {e 6 Met*(X) e is locally qualitatively saturated if whenever q is a *-metric (Vx e X, Ve > 0,3p e T>, 3<5 > 0, Vy e X : p(x, y) < 6 => q(x, y) < e) =» q e T>.

54

(UQL)

COLEBUNDERS ET AL.

P is uniformly qualitatively saturated if whenever q is a *-metric (Ve > 0,3p e V, 3(5 > 0, Vz, y & X : p(x, y) < 5 =>• g(x, y) < e) =>
(LQN)

P is locally quantitatively saturated if whenever q is a *-metric (Vx e X,Ve > 0, Vw < oo, 3p € P, Vy e X : g(or, y) A w < p(z, y) + e) => q £ P.

(UQN)

P is uniformly quantitatively saturated if whenever q is a *-metric (Ve > 0 , V w < oo,3p€ P, Vx, y £ X : q(x, y) A w < p(z, y) + e) => g <E P.

4. Examples and counterexamples 4.1. Local qualified ideal: PrTop, Top, CrPrTop, Greg 4.1.1. PrTop, [1, 5, 6, 10, 14, 25], is the construct of pretopological spaces and continuous maps. Define PrTop9pm to be the construct whose objects are sets equipped with a nonempty downset T> of quasi-pre-metrics satisfying (IDL) and (LQL) and with morphisms those / : (X,P) -» (X',P') satisfying d! & V => d' o / x / e P. Let G : PrTop -> PrTop9pm be the concrete functor defined on objects by G(X_) = (X, T>) where P = {P I P quasi-pre-metric , (X, Tp) < X}. That G is indeed a functor follows from the observation that when / : X_ —> 2C' is a continuous map and G(2Q = (X,P), G(2C') = (X',P') then q e P' implies Bqofxf(x,e) = f-lBq(f(x),e)

Vx&X,Ve>0.

(1)

Hence (X',Tq) < X_' implies (X,Zj0/x/) ). For p and p' in P observe that -Bpvp' (z, e) = Bp(x, e) n Bp- (x, e)

Vz e X, Ve > 0.

So clearly p Vp' e P which implies G(2C) satisfies (IDL). With the same notations, suppose Vx e X, Ve > 0,3p € P, 3<5 > 0 such that p(x, y) < 6 =>•

METRICS

55

q(x,y) < e, then (X,TP) < X_ implies (X,Tq) < X_ and so we can conclude q 6 T>. Next we define F : PrTop9pm —> PrTop the concrete functor where for (X, D), with 23 a downset of quasi- pre- metrics satisfying (IDL) and (LQL) we put d€T>

the supremum being taken in PrTop. That F is a concrete functor follows from the observation (1) above. Next we prove that F and G are mutually inverse isomorphisms. Let X_ be a pretopological space. For A, V C X with V a neighborhood of A put P(A,V) '•= I-AX(X\V)- Then P(A,V) is a quasi-pre-metric on X and (X, TP(A v ) ) < X_. If V is any neighborhood of x in X_ then V clearly is a neighborhood of x in (X,Tp({x} v)). Hence X=

sup V nbhd of A

(X,TP

).

This allows us to conclude that F o G = lprTopLet (X,T>) be a PrTop9pm-object. Clearly G o F(X, V) > ( X , D ) . To show the other inequality, let G(F(X, V)) = (X,P) and let q be a quasipre-metric such that q e P i.e pf,7^) < X_. Further, let x € X and let e > 0, then it follows that Bq(x,e) is a neighborhood of x in X. So there exist pi , . . . , pn e D and 5 > 0 such that

If we now put p := p\ V • • • Vpn then Bp(x, 6) C Bq(x, e) and it follows that q e T> by (IDL) and (LQL). 4.1.2. Top is the construct of topological spaces and continuous maps. Define Top9"* to be the construct whose objects are sets equipped with a nonempty downset T> of quasi-metrics satisfying (IDL) and (LQL) and with morphisms those / : (X, T>) -» (X', T>') satisfying d' e £>' => d' o f x / e T>. Let G : Top —» Top9Tn be the concrete functor defined on objects by G(2Q = (X, V) where T>= { p \ p quasi-metric , (X, Tp] < X}. That G is indeed a functor and that the objects in the codomain of G satisfy (IDL) and (LQL) is proved in exactly the same way as for PrTop.

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COLEBUNDERS ET AL.

F : Top9 —> Top is the concrete functor defined on objects by

where the supremum is taken in Top. In order to show that FoG = ITOP observe that every topological space can be written as X_ = supA O pen(-^'^p/i) where we have used the notation PA = P(A,v) when A = V, and where all the PA'S clearly are quasi- metrics. In order to show that GoF = ITOP exactly the same procedure as for PrTop applied to quasi- metrics instead of quasi-pre- metrics can be applied. 4.1.3. We define CrPrTop to be the concretely reflective hull of all (X, Tp) where p is a pre-metric on X. So by definition, CrPrTop is concretely reflective in PrTop. The objects of CrPrTop are called completely regular pretopological spaces. CrPrToppm is the construct whose objects are sets equipped with a nonempty downset T> of pre-metrics satisfying (IDL) and (LQL) and with morphisms those / : (X,T>) -> (X',W) satisfying d' e V =» d' o f x / € V. For a CrPrTop object X define GQQ = (X,V) where X > = { p \ p pre-metric , (X, Tp) < X_}For (X, V) an object of CrPrToppm define

where the supremum is taken in CrPrTop, (which coincides with the supremum in PrTop). Observe that when X_ is in CrPrTop, there is an initial source in PrTop, (fa : X_ —» (Yi,Tpi))i&j. This also means that

X_ =

sup(X,Tpi0fiXfi). t€l

Using this fact, and changing quasi-pre-metric in the proof for PrTop, to pre-metric one can conclude that the functors F and G are mutually inverse. 4.1.4. Creg is the construct of completely regular topological spaces, which is known to be the concretely reflective hull of the class of metrizable topological spaces. Moreover, every completely regular space X_ can be written as

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for T> some class of metrics on X. Using the same procedures as before one shows that Creg is isomorphic to Cregm, the construct with objects, sets equipped with a nonempty downset V of metrics satisfying (IDL) and (LQL) and with morphisms those / : (X,T>) -> (X',T>') satisfying d' € V =» d' o f x / e T>. 4.2. Local qualified: PrCl, Cl,CrCl In this section we investigate modified versions of the categories in the previous section, by dropping one of the saturation conditions, namely (IDL). Define PrCl9pm, the construct whose objects are sets structured by a nonempty downset V of quasi-pre-metrics, satisfying (LQL) and with morphisms those / : (X, D) -> (X', V) satisfying d' G £>' => d' o / x / 6 T>. In the same way C\qm (and CrClm) have as objects sets structured by a nonempty downset of quasi-metrics (metrics) satisfying (LQL) and have morphisms defined analogously. These constructs are isomorphic to some well known topological constructs of (pre-) closure spaces in the sense of Birkhoff [3]. Cl is the construct of sets X, structured with some closure cl satisfying the following properties c!0 = 0, A C cl(A), A c B implies d(A) C d(B), cl(cl(A)) = cl(A), for A C X, and with morphisms the continuous maps, i.e. those / from X to X' satisfying f(d(A)) C d ' ( f ( A ) ) for subsets A of X. PrCl is the larger construct obtained by dropping the condition of idempotency. In [7] PrCl was obtained as extensional topological hull of Cl. Finally CrCl is the concretely reflective hull in Cl of the class of all metrizable topological spaces. Note that this class appears as the class of closure spaces that are "uniformizable" by some uniform pre-nearness structure, [12], (see section 4.4 and the table at the end). Using formally the same functors as those described in section 4.1, PrCl (Cl and CrCl) are shown to be isomorphic to PrCl9pm (Cl9m and CrClm) by essentially the same proofs as those developed in the previous section. The fact that initial structures in PrCl (in Cl and CrCl) are formed differently compared to the pretopological case is responsible for the new form of saturation in which (IDL) does not play any role. 4.3.

Uniform qualified ideal: sqUnif, sUnif, qUnif, Unif, Prox

We apply the terminology used for instance in [1, 4, 14, 15, 18, 25] . A semiquasi uniform structure U on X is a filter on X x X satisfying the only

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condition that all members contain the diagonal. If moreover for all U € U there exists a V e U such that V o V C U then the structure is a quasi uniformity. If a semi-quasi uniformity U satisfies U e U => U~l £ U then U is a semi uniformity and if both conditions are fulfilled U is a uniformity. We consider the constructs sqUnif, sUnif , qUnif , Unif whose objects are sets structured with a semi-quasi uniform structure, a semi uniform structure, a quasi uniform structure or a uniform structure respectively, and where the morphisms are the uniformly continuous maps. It is well known [15] that the construct Unif is concretely isomorphic to the construct Unif m , whose objects are sets structured with a collection V of metrics satisfying (IDL) and (UQL) and with morphisms those / : (X, V) -> (X', V) satisfying d' e V ^ d' o / x / e V. The classical proof can easily be generalized to the more general cases mentioned above. qUnif m , (sUnifpm, sqUnif9pm) are the constructs whose objects are sets structured with a collection of quasi-metrics (premetrics, quasi-pre-metrics) satisfying (IDL) and (UQL) and where morphisms are again defined in the same way. The isomorphisms between qUnif and qUnif9"1 (between sUnif and sUnif pm , and between sqUnif and sqUnif 9pm) are constructed similarly as in the uniform case. For instance, in the case of sqUnif, one uses the concrete functors G and F defined as follows. Let G : sqUnif —-> sqUnif9pm be the concrete functor defined on objects by G(X_) = (X, T>) where V = {p\p quasi- pre- metric , (X,UP) < X}, where Up — stack{Vp j r > 0} and where V£ = {(x, y) \ p(x, y) < r} Next we define F : sqUnif9*"71 —> sqUnif the concrete functor where for (X,T>), with V a downset of quasi-pre-metrics satisfying (IDL) and (UQL) we put

the supremum being taken in sqUnif. Prox is the construct of proximity spaces and proximal maps [1, 2, 16, 23]. It is well-known that Prox is isomorphic to the full subconstruct of Unif consisting of all totally bounded uniform spaces, which is concretely reflective in Unif. Define Proxttm as the construct whose objects are sets equipped with a nonempty downset V of totally bounded metrics satisfying (IDL) and (UQL) and with morphisms those / : (X,T>) -> (X',V) satisfying d' & T>' =>• d' o / x / e T>.

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Let G : Prox —> Proxt6m be the concrete functor denned on objects by G(X) = (X, V) where V = { p \ p totally bounded metric , (X,UP) < X_}. Next we define F : Proxtbm —> Prox the concrete functor where for (X, T>), with T> a downset of totally bounded metrics satisfying (IDL) and (UQL) we put

F(X,V} = sup(X,Ud), the supremum being taken in Prox, which coincides with the supremum taken in Unif. The proof that GoF = 1 goes exactly as for Unif. To see that FoG = 1, let (X, U) be a totally bounded uniform space. Then in Unif we have

(X,U) = sup (X,Ud). UdCU

Note that every d in the collection is totally bounded and therefore the equality also holds in Prox.

4.4. Uniform qualified: UPrNear Next we investigate a situation similar to the closure case. Dropping (IDL) in the description of Topqm gave rise to C\qm and its isomorphic construct Cl. We start with Unif m and again we drop the condition (IDL). We obtain the construct UPrNearm, whose objects are sets equipped with a nonempty downset T> of metrics satisfying (UQL) and with morphisms those / : (X,T>) -» (X',V) satisfying d1 £ D1 => d' o / x / e X>. In fact this category is concretely isomorphic to the construct UPrNear which is a full subconstruct of PrNear, the category of pre-nearness spaces and uniformly continuous maps as introduced in [16]. Objects in UPrNear are prenearness spaces described by collections of covers satisfying the ^-refinement axiom. UPrNear plays an important role in the extension theory for closure spaces. The proof of the isomorphism between UPrNear and UPrNearm is based on formally the same functors as those in the section on Unif. The fact that (IDL) does not play any role in this case, again is a direct consequence of the description of the initial structures.

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4.5. Local quantified ideal: PrAp, Ap, UAp The categories PrAp, Ap and UAp were introduced and studied in [9, 14, 19, 20, 21]. PrAp is the construct whose objects are sets structured by a function 5 in the following way. Given a set X denote the set of all subsets of X by 2 X . A function 6 : X x 2X —> [0, oo]

is called a pre-approach distance (on X) if it satisfies the following properties: (Dl) Vz 6 X • 6 (x, {x}) = 0, (D2) V z € X : < 5 ( : r , 0 ) = oo, (D3) Vz € X, VA, B e 2X : 6 (x, A\J B] = min (6 (x, A) , 6 (x, B)). Morphisms in PrAp are contractions. If (X, 6) and (X1, 6') are approach spaces, then a function / : X —> X' is called a contraction if Vx 6 X, VA € 2X : 6' (f (x) J(A))<5 (x, A) . The construct Ap is the full subconstruct whose objects satisfy a supplementary condition (D4) denned below. For any subset A C X and any e € [0, oo], we define A(s} :={xeX\ 5(x,A) <e}. Then 6 is called an approach distance (on X) if it moreover satisfies (D4) V z e X , W l e 2 - v , V £ € [O.oo] :S(x,A) <6(x,A^)+£. UAp is the approach version of the construct of completely regular spaces. An approach space (X, 5) is metrizable if there is a metric d on X such that

8(x, A) = inf d(x, a) for A C X, x 6 X. UAp is the concretely reflective hull in Ap of all metrizable approach spaces. It was shown in [11, 19] that all three categories have isomorphic descriptions PrAp9pm, Ap9™, UApm as constructs whose objects are sets equipped with a nonempty downset T> of quasi-pre-metrics in the case of PrAp, of quasi-metrics in the case of Ap and of metrics in the case of UAp, satisfying (IDL) and (LQN) and with morphisms those / : (X, T>) —» (X', V) satisfying d1 e £>' => d' o / x / e T>.

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4.6. Uniform quantified ideal: qUG, UG, EfGap The constructs qUG, resp. UG, of quasi uniform gauge spaces were intoduced in [22] as quantified counterparts to the constructs qUnif, resp. Unif. The objects of qUG consist of a set X with as structure a downset F, of quasi-pre-metrics satisfying the saturation conditions (IDL) and (UQN) together with a supplementary condition: (T) VT e r, Vw < oo, 3^ e T, Vi,y, z e X \ 7(1, z) < i/)(x, y) + ip(y, z). Using a quantified version of Urysohn's metrization lemma, it was then shown in [26] that condition (T) guarantees that there is a basis for F consisting of quasi-metrics. If furthermore the following condition

(S) V7 e r :7-: e r,

(where 7~1(x, y) :— ~/(y,x) for all x,y £ X) is fulfilled, then there is even a basis consisting of metrics. The morphisms in qUG are the uniform contractions in the following sense: for quasi uniform gauge spaces (X, F) and (X1, F'), a function / : (X, F) —» ( X ' , F') is called a uniform contraction if 7' o / x / e F for all 7' 6 F'. From this it is deduced in a similar way as for Ap that qUG, resp. UG, are isomorphic to constructs qUG?m, resp. UG m , the objects of which are sets structured by a nonemty downset of quasi-metrics, resp. metrics, satisfying the conditions (IDL) and (UQN), and where again in both cases, the morphisms are as before. A quasi uniform gauge space (X, F) is called (quasi-)metrizable if there exists a (quasi-)metric d on X for which F = {7 e [0, oo}xxx | Vi,y e X : 7 (x,y) < d(x, y)} =: Td. qUG, resp. UG, equal the concretely reflective hull in qUG of the class of all quasi metrizable, resp. all metrizable, objects. In [13], the authors introduced and examined the construct EfGap of Efremovic gap spaces, which serve as the quantitative counterpart to the construct Prox in approach theory. The objects of EfGap are sets X which are structured by a gap functional g : 2X x 2X -» [0, oo]

satisfying the following axioms: (Gl) (G2) (G3) (G4) (G5)

VA,B e 2* : g(A,B) = 3(B,A), VA, B e 2X : 0 e {A, B} => g(A, B) = oo, VA, B G 2X : A n B ^ 0 =* g(A, B) = 0, Wl, B, C e 2* : g(A, BuC)= g(A, B) A Q(A, C), VA B, C e 2* : 9(A, B) < S(A, C) + supc6C S({c}, B).

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COLEBUNDERS ET AL.

(G6) VA,B e 2 x ,Ve,r?eR+: ( V C € 2 x : g ( A C ) < £ or 9(X \ C, B) < r,) =* 3(A,B) < e + r,. For Efremovic gap spaces (X, 0), (X',Q') the EfGap-morphisms between them are those functions / : (X, 0) —» (X1, 0') which are proximally contractive in the sense that

VA,B e 2X : fl'(/(A),/(B)) < fl(A, J3). Let us also recall from [22] that Unif can be concretely embedded into UG as a concretely bireflective and bicoreflective full subconstruct and that the coreflector U : UG —> Unif acts as the 'uniform underlying functor' in uniform approach theory. For a uniform gauge space X_ = (X, F), t/(2Q is denned to be the uniform space on X having

{{(x,y) € X x X | 7(x,y) < a} | 7 & T,a > 0} as a base for the entourage filter. It was consequently shown by the authors in [13] that the construct EfGap is isomorphic to the full subconstruct tbUG of UG, formed by those uniform gauge spaces X_ for which U(X_) is a totally bounded uniform space. Therefore, in order to show that the objects and morphisms of EfGap can be described in the desired way using nonempty downsets of totally bounded metrics which satisfy the supplementary conditions (IDL) and (UQN), we can as well show that the same conclusion holds for tbUG. So define tbUGtf>m to be the construct whose objects are sets structured by a nonempty downset of totally bounded metrics satisfying the conditions (IDL) and (UQN) and with morphisms those functions / : (X, T>] —> (X',T>') for which Vd' € & : df o / x / e V.

Define a concrete functor G : tbUG -> tbUG tf>m , by putting for each tbUG-object X_, G(X_) = (X, T>] where X> = {p | p totally bounded metric, (X, Fp) < X}. We also define F : tbUG*bm —> tbUG to be the concrete functor, completely determined by putting

for each tbUGti>m-object (X,T>), where the supremum is taken in tbUG (which is the same as taking it in UG, since the former is a concretely

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bireflective subconstruct of the latter). The fact that G o F = 1 is proved exactly in the same way as for UG. In order to see that F o G = 1, let (X, F) be a uniform gauge space such that U(X, F) is a totally bounded uniform space. We already know that in UG the equality

holds. Since for each metric d on X with F^ c F, automatically (X,Ud) — U(X, Fd) < U(X, F), all the metrics arising in the supremum are necessarily totally bounded, finishing the argument.

4.7. Principal ideal: Metqp, Metp, Met9, Met Obviously Met9p, Metp, Met9, Met are all isomorphic to respectively Met9pm, Metpm, Met9"1, Metm where in all cases the objects are sets endowed with a principal downset of type T>d = {e * —metric | e < d} for d a *-metric.

4.8. Non-examples Some well-known topological constructs do not fit in the table at the end since the cardinality of their fibres is too large. This is the case when some of the constructs in the table are replaced by bigger ones. For instance when PrTop is enlarged to its cartesian closed topological hull PsTop [1, 6, 25] consisting of pseudotopological spaces there does not exist any saturation inducing an isomorphic description of PsTop by means of sets structured by collections of generalized metrics. The same happens when Unif is enlarged to Near [16], the construct of nearness spaces or to Ulim [25] , the construct of uniform limit spaces, or to Chy [8] , the construct of Cauchy spaces. We now develop an argument that can be used in the case of Near, Chy and Ulim. Take N as underlying set. Choosing any collection A of nonprincipal ultrafilters on N we define a merotopic structure HA on N in the following way. For a cover V of N we put V 6 HA iff V intersects F whenever F £ A. That HA satisfies the axioms of a merotopy is easily seen. Observe that the associated interior operator int^ on subsets Y of N satisfies intMA (Y) = Y. It follows that HA is a nearness structure. The small collections of this structure are exactly those B for which stackS contains either a principal ultrafilter or one of the ultrafilters in A. It follows that in fact HA is a Cauchy structure and therefore also a uniform limit structure. The

COLEBUNDERS ET AL.

64

observation about the small collections moreover allows us to conclude that the structures HA are in one-one correspondence with the collections A. Therefore the number of nearness structures (uniform limit structures) 2 0 on N is at least 22 . Near (Ulirn, Chy) cannot have a concretely isomorphic description as a construct whose objects are sets structured by certain collections of generalized metrics. Indeed the number of quasi-premetrics on N equals 2N° and hence there are at most 22"0 structures on N consisting of collections of generalized metrics. The example can be modified in such a way that the structure HA becomes a pseudotopological structure. Therefore the same argument applies also to Pstop. N

4.9. Summary The following table summarizes the previous results. The constructs represented by squares in the table are completely determined by their row (sat) and column (*). However to the best of our knowledge, these constructs have not been introduced in the literature and hence we have refrained from naming them. sat / * LQL + IDL LQL UQL + IDL UQL LQN + IDL UQN + IDL PRIDL

|| qpm

PrTop PrCl sqUnif D PrAp D Met9P

pm CrPrTop D sUnif D D D Metp

qm

m

tbm

Top Cl qUnif D Ap qUG Met9

Greg CrCl Unif UPrNear UAp UG Met

D D Prox D D EfGap D

Bibliography [1] Adamek J., Herrlich H. and Strecker G.E., Abstract and concrete categories,John Wiley, New York, 1990 [2] Bentley L., Herrlich H., Husek M., The historical development of Uniform, Proximal and Nearness concepts in Topology, Handbook of the History of General Topology, Vol. 2 Kluwer Ac. Publ. (1998), 577-629 [3] Birkhoff G., Lattice Theory, A.M.S., Providence, R.I., 1940 [4] Carlson J.W., Nearness and uniform type structures, Portugaliae Mathematica 40 (1981), 469-479 [5] Cech E., Topological Spaces, Publ. Czechoslovak Acad. Sci., Interscience, John Wiley, 1966

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[6] Choquet G., Convergences, Ann. Univ. Grenoble, Sect. Math. Phys. 23 (1948), 57-112 [7] Claes V., Sonck G., The quasitopos hull of the category of closure spaces, Appl. Gen. Top., to appear [8] Lowen-Colebunders E., Function classes of Cauchy continuous maps, vol. 123 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1989 [9] Lowen-Colebunders E., Lowen R., Topological quasitopos hulls of categories containing topological and metric objects, Cahiers Topol. Geom. Diff. Categ. 30 (1989), 213-228. [10] Lowen-Colebunders E., Lowen R., Supercategories of TOP and the inevitable emergence of Topological Constructs Handbook of the History of General Topology, Vol. 3 Kluwer Ac. Publ. (2001), 969-1026 [11] Lowen-Colebunders E., Lowen R. and Verbeeck F., Exponential objects in the construct Prap, Cahiers Topol. Geom. Diff. Categ. 38 (1997) 259-276 [12] Deses D., De Groot-Van der Voorde A., Lowen-Colebunders E., Extensions of closure spaces, Appl. Gen. Top., to appear [13] Di Maio G., Lowen R., Naimpally S.A., Sioen M., Gap spaces : gapcizing proximities, submitted [14] Dikranjan D., Tholen W., Categorical structure of closure operators, Math. Appl., Kluwer Ac. Publ., 1995 [15] Gillman L., Jerison M., Rings of continuous functions, Springer Verlag 1976 [16] Herrlich H., Topological Stuctures, Math. Centre Tracts 52 (1974), 59-122 [17] Herrlich H., Strecker G.E., Category Theory, Allyn and Bacon 1973 [18] Kiinzi H.-P. A., Nonsymmetric distances and their associated topologies, Handbook of the History of General Topology, Vol. 3 Kluwer Ac. Publ. (2001), 853-968 [19] Lowen R., Approach Spaces: the Missing Link in the Topology Uniformity Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997 [20] Lowen R., An Ascoli theorem in approach theory, Topology Appl. to appear [21] Lowen R., Sioen M., A unified functional look at completion in Met, Unif and Ap, Appl. Categ. Struct., 8 (2000), 447-461 [22] Lowen R., Windels B., AUnif: a common supercategory of pMet and Unif, Internat. J. Math. Math. Sci 21 (1998), 1-18 [23] Naimpally S.A., Warrack B.D., Proximity spaces, Cambridge University Press, 1970 [24] Preuss G., Theory of Topological Structures, Math. & Appl., Kluwer Ac. Publ., 1988 [25] Preuss G., Foundations of Topology : an approach to Convenient Topology, Kluwer Ac. Publ., 2002 [26] Windels B., Uniform Approach Theory, Ph.D. Thesis, University of Antwerp, 1997

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A UNIFORM SPACE PROOF OF A METRISATION THEOREM

P. J. COLLINS St Edmund Hall Oxford, 0X1 4AR, UK. E-mail : [email protected] We give a short, elementary proof of the metrisability of a compact Hausdorff with Gj diagonal, using uniform space theory and in the spirit of the work of Horst Herrlich. Mathematics Subject Classifications (2000): 54E35,54E15 Keywords: metrization theorem, G^-diagonal, uniform spaces

Horst Herrlich has shown us, in so many of his papers, how the right mathematical language can give us greater understanding of classical theory and classical results. Of course, in exposing a classical result as following naturally and easily from basic theory, a lot depends on what one considers 'basic theory'. Here we argue, as in an earlier paper (Collins [3]), for greater acceptance of the theory of uniform spaces as part of every topologist's basic tool-bag. The theorem we aim at is the following well-known and elegant, yet elementary, result. Theorem 1 A necessary and sufficient condition for a compact Hausdorff space X to be metrisable is that the diagonal A of X x X is a G&. The proof we give, rather shorter than may be found, for example, on pages 196-197 of J.-I. Nagata [4], turns on the next two results from the basic theory of uniform spaces. Nagata's in [4] relied on an application of the topological analogue of the Alexandroff-Urysohn Theorem, which is the first of them. Theorem 2 (see Bourbaki [2], page 152). A Hausdorff uniform space is metrisable if and only if its uniformity has a countable basis. 67

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Theorem 3 (see Bourbaki [1], page 199). There is a (unique) uniformity compatible with the topology of a compact Hausdorff space, and all the neighbourhoods of the diagonal G$ of X x X are entourages in this uniformity. Proof. Theorem 1 We first recall that the family of all open entourages forms a basis for the uniformity of X as also does the family of all closed entourages of X. Necessity If X is metrisable, then the space X has a countable basis for its uniformity, the intersection of the interiors of elements of which is A (since X is Hausdorff). Sufficiency If A = f| [/» where each Ui is open, then, using Theorem 3, there exists a decreasing sequence of closed entourages C = {d} with intersection A. If C is not a basis for the uniformity on X, there exists an open entourage V such that

is a closed filter-basis on X x X. Since X x X is compact, there is a cluster point z of J7. But then both

z 0 V and z 6 nCi = A This is impossible, and hence X is metrisable by Theorem 2.

D

References [I] N. Bourbaki, General Topology, Part 1, (Hermann and Addison- Wesley), 1966, chapter II. [2] N. Bourbaki, General Topology, Part 2, (Hermann and Addison- Wesley) , 1966, chapter IX. [3] P.J. Collins, On uniform connection properties, Bull.Amer.Math.Soc. vol.78, no.4 (1971), 372 - 374 [4] J. Nagata, Modern General Topology (North- Holland), 1968, chapter VI.

CLASSIFICATION OF CLOSURE OPERATORS FOR CATEGORIES OF TOPOLOGICAL SPACES*

DIKRAN DIKRANJAN Dipartimento di Matematica e Informatica Universita di Udine Via delle Scienze 206 33100 Udine, Italy E-mail: dikranjan@dimi. uniud. it

WALTER THOLEN AND STEPHEN WATSON Department of Mathematics and Statistics York University 4700 Ke.de Street Toronto ON, Canada M3J 1P3 E-mail: [email protected] E-mail: [email protected]

We describe some features of the (very) large lattice of all closure operators of the category Top of topological spaces and exhibit the central role of the Kuratowski closure, as the only proper hereditary closure operator which behaves well on products. Additivity alone characterizes the Kuratowski closure in some subcategories of Top, such as all connected and locally connected metrizable spaces or all zero-dimensional metrizable spaces. Mathematics Subject Classifications (2000): Primary 18A20, 18B30, 54B30, 54D25; Secondary 54A10, 54B17, 54D10, 54D15, 54D25, 54D30 Keywords: closure operator, (^-closure, sequential closure, metrizable space, Alexandroff-Tucker space

1. Introduction

Ever since the appearance of Salbany's paper [13] closure operators have become a standard tool in categorical topology, predominantly in the study 'This work has been supported by the Natural Sciences and Engineering Research Council of Canada and the Ministry of Research and University of Italy 69

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of epimorphisms of full subcategories of the category Top of topological spaces, via their so called regular closures (namely, the closure operators introduced by Salbany). There is therefore a profusion of closure operators on Top (at least as many as there are subclasses of a proper class), especially since general Dikranjan-Giuli closure operators are not even required to be idernpotent (all regular operators are idempotent). By definition, a categorical closure operator c is given by a coherent family of closure operations ex, one for every space X. The main purpose of this paper is to show that, in the presence of good properties of GX, such as additivity (cx(M U N) = cx(M) U cx(N) for all M, N C X) or hereditariness (cy(M) = cx(M) n Y for M C Y C X), the values of ex for very special spaces X determine the values of cx for all other spaces X. Of particular importance are the cases when X is the Sierpiriski dyad or the real line. Among other things these considerations lead us to a characterization of the usual Kuratowski closure operator of Top, as the only proper additive and hereditary closure operator with the so called Tychonoff property. In certain subcategories of Top, additivity alone suffices to characterize the Kuratowski closure. In none of these characterizations idempotency plays an explicit role; rather, it comes "for free". Along the way we exhibit many known and new closure operators of Top which shed further light on its "lattice" of closure operators. Here is a brief outline of the contents of the paper. In Section 2 we give the definition of closure operator and several examples. In Section 3 we study three properties of closure operators and the interrelations between them: idempotency, productivity and hereditariness. Section 4 is devoted to various degrees of additivity, the lowest one being the usual Cech additivity property. Then we describe closure operators solely depending on its effect on "simple" spaces such as Sierpifiski's dyad or the real line R. In Section 5 we "test" Sierpifiski's dyad, which helps us to describe all fully additive hereditary closure operators of Top and, in particular, all Cech hereditary closure operators of the category FinTop of finite topological spaces. Let us note that full additivity together with hereditariness presents a rather strong restriction on the closure operator. There are many closure operators on finite spaces; for example there are four Cech clops which are distinct on the Sierpinski dyad while only three Cech closure operators are distinct on the real line. In Section 5.2 we classify closure operators on finite spaces. More on closure operators on finite spaces can be found in the forthcoming paper [9]. In Section 5.2 we continue the study of hereditary closure op-

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erators. Here we discuss the Tychonoff property (formerly called the finite structure property for products, or briefly f.s.p.p. [6]), naturally related to products, and we show that the Kuratowski closure is the only hereditary Cech closure operator with the Tychonoff property on Top. In Section 6 we trade Sierpiriski's dyad for a richer space, namely the real line R. This permits us to obtain a classification of all Cech closure operators into three classes: R-fine, R-coarse, and R-tame closure operators. Finally, in Section 7, we use this classification to characterize the Kuratowski closure among Cech closure operators on metrizable-like spaces. We show here that locally "good" spaces admit few closure operators, in fact only the Kuratowski closure in most cases. It should be stressed that now the good properties of the spaces under consideration permit us to relax the strong restraints on the closure operators used earlier. Actually, most often Cech additivity is sufficient. The following full subcategories of Top will be discussed in this note: Topo (Tb-spaces), Topi (Ti-spaces), Tych (Tychonoff spaces), FU (Frechet-Urysohn spaces), Metr (metrizable spaces). 2. The large lattice of closure operators of Top and its subcategories Let A be a full subcategory of Top, the category of topological spaces and continuous maps, which contains a singleton space 1 and which is closed under subspaces (so that Y C X e A implies Y € A). Definition 2.1. (Dikranjan and Giuli [4]) A closure operator c of A is given by a family of maps cx : 2X —> 2X, X & A, such that for every X in A • cx is extensive : M C cx(M) for all M e 2 X , • cx is monotone : cx(M) C cx(N) for all M C N C X, and for every continuous map / : X —» Y in A • f is c-continuous : f(cx(M)}

C cy(/(M)) for all M

CX.

The conglomerate CL(A) of all closure operators in Top is partially ordered by

c < d «=> cx(M) C dx(M) for all M C X e A; one says that c is finer than d, or that d is coarser than c in this case. Arbitrary meets and joins of families of closure operators exist in CL(A),

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and they are given by set-theoretic intersections and unions, respectively; we use the lattice-theoretic notations A, /\, V, VNot surprisingly, the most important closure operator on Top is given by the usual Kuratowksi closure k\(M) = M of a subset M in X. The discrete operator j with jx(M) = M for all M C X e A is the finest closure operator, and the trivial operator t with tx (M) = X for all M C X e A is the coarsest closure operator. We note that ix(0) ^ 0 unless X = 0; however, for every other closure operator c of A one has c\ (0) = 0 for all X € A. (Exploit the continuity condition for maps X —» 1 and 1 —> X to see that cx(0) = 0 or cx($) = -X for all X £ A, depending on whether ci(0) = 0 or ci(0) = 1 for a singleton space 1.) The trivial closure operator t has an immediate predecessor, the indiscrete closure operator g, given by 5x (0) = 0 and gx(M) = X for each X & Top and 0 ^ M C X € A These names are motivated by the fact that a space X is discrete (indiscrete) precisely when kx = jx (kx = 9x, respectively). Any closure operator different from j, g, t is called proper on A. We say that c is proper on a space X <E A if ex 7^ tx, gx, jx- Next we shall show that there is a finest and a coarsest proper closure operator of Top. Recall that the quasi-component qx(M] of M C X in X e Top is the intersection of all clopen (=closed and open) sets of X containing M. "Dually" define the minimal proper closure operator nx(M) to be the union of all indiscrete sets of X which meet M. It is easy to check that q = {lx}xeTop and /j, = {^x}xeToP are proper closure operators of Top (see also 2.5 below). Moreover: Proposition 2.1. For every proper closure operator c of Top one has j < ^ 2 with M = X~M({^}))- Hence we must have c-2 = g-z- Now consider y e nx(M), for any space X and M C X. Then y & IJLX({X}} for some x £ M. The subspace Y — {x,y} of X is indiscrete. Hence y G CY({X}) C cx({x}) C /j,x(M). This proves C c x (M).

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For the proof of c < q one provides 2 with the discrete topology and proceeds "dually". D When dealing with closure operators of a subcategory A it is often convenient to know that they may be extended to Top. This is always possible when A is epireflective in Top, i. e., when A is not only closed under subspaces, but also under direct products. In fact, given c £ CL(A), for any space X, let TX : X —> RX be the reflection into A. Denoting by M the range of the map R(M <-> X) : RM —> RX, we may define

(1) One can then easily prove (cf. Dikranjan-Tholen [6]): Proposition 2.2. If A is an epireflective subcategory of Top, formula (1) gives for every closure operator c of A an extension to Top. This way, CL(A) gets reflectively embedded into CL(Top), with the reflector given by restriction. QED In case A = Top0 is the category of To-spaces, one has a kind of converse assertion (see Dikranjan, Giuli and Tholen [5]): Proposition 2.3. Every non-discrete closure operator c of Top is an extension of a closure operator of Top0, i. e. satisfies formula (1) with rx '• X —> RX the T0-reflection of a space X. QED 3. Idenipotency, hereditariness and productivity Definition 3.1. Let c be a closure operator of a full and (for simplicity) epireflective subcategory A of Top, and let K be a cardinal number. One calls c • idempotent at X e A if cx(cx(M}) = cx(M) for all M C X, • hereditary at X 6 A if c y (M) = cx(M) n F for all M C F C X. Furthermore, c is idempotent (hereditary) if it is idempotent (hereditary, resp.) at every X e A, and c is called • K-productive if cx(Yli€lMi) = Yl^jCx^Mi) holds for all M, C Xi e A, i e /, cardl < K, with X = fl^/ Xi. Finally, c is • finitely productive if it is ^-productive, and

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• productive if it is K-productive for every K. Each of the properties of idempotency, hereditariness and K-productivity is stable under meet in CL(A); hereditariness is also stable under join in CL(A). Consequently, for each closure operator c one can find a finest idempotent (hereditary) closure operator in CL(A) coarser than c, called the idempotent (hereditary) hull of c and denoted by c°° (che, resp.). Similarly, there is a coarsest hereditary closure operator in CL(A) finer than c, called the hereditary core of c and denoted by Che- Of these hulls and cores, in general only c°° seems to be computationally accessible. One way is to consider the ordinal powers ca (recursively defined by c° = j, ca+l = cca and ca = V{c" : /? < a} for a limit a); then c°° is the join of all ca. Another way is discussed in Remark 3.2 below. For c,d£ CL(Top) one defines the composite dc and the cocomposite d * c as follows. For X e Top and a subspace M of X one puts

(dc)x(M) = dx(cx(M)} and (d*c)x(M) = dcx(M)(M). The verification of the fact that dc and d*c are closure operators is straightforward. The discrete closure operator j is neutral with respect to composition and acts like a zero with respect to cocomposition, while for the trivial operator t these roles get interchanged:

cj = jc = c, c * j = j * c — j, ct — tc = t, c*t = t*c = c hold for every c e CL(Top). The following theorem gives some surprising interrelationships between the properties introduced in 3.1. It has been proved in a more general categorical context in Dikranjan-Tholen [6]; here we give ad-hoc proofs for the reader's convenience. Lemma 3.1. For c, d G CL(A), if the composite cd is hereditary, then cd = c V d. Proof. One always has cVd < cd. Now suppose that for some M C X 6 A there is a point x € c\(d\(M^) which is not in cx(M) U dx(M). For the subspace Y = M U {x} of X one then has x g CY (M) U dy (M) C (cx (M) U dx(M))r\Y. Hence c y (M) = M = dY(M) and then M = cy(dy(M)). On the other hand Y C cx(dx((M}}, so that the composition cd must fail to be hereditary. D

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Corollary 3.1. If c,d € CL(A) and cd ^ c V d, then there exist M C Y C X & A such that M is cd-dosed in Y, but the cd-closure of M in X contains Y; in particular, cd is not hereditary. A closure operator c of A has order a if a is the least ordinal with cQ+1 — ca; if no such a exists, the order is oo. We denote the order of c by o(c). Note that co(c) = c°°. Theorem 3.1. Let c 6 CL(Top). (a) If c is not idempotent, then the powers ca (2 < a < o(c)) are not hereditary. (b) If c has infinite order o(c), then for every limit ordinal a < o(c) the power ca is not a+ -productive; in particular, if o(c) is a limit ordinal, then c°° is not productive. (c) c" is finitely productive. (d) If c is idempotent (i. e., if o(c) < 1), then c is finitely productive. (e) If c is finitely productive, then every power ca (0 < a < oo) is finitely productive. Proof, (a) We may assume that c ^ g, t and a < o(c). Fix any 1 < (3 < a and apply Corollary 3.1 to the closure operators c and c@ with cc@ — c@+l > c® = c\l c® to conclude that c@+l is not hereditary. This proves item (a) in case a = /?+ 1 is a successor. In case a is a limit and (3 < a, apply Corollary 3.1 to the closure operators c and c13 to get Mp C Yp C Xp e A such that Mp is cc^-closed (and, hence c-closed) in Yp and the cc^-closure of Mp in Xp contains Yp. Now let X = Y[p Xp be the coproduct of the spaces Xp, Y = JJ^ Yp and M = HgMp. We shall show that M is enclosed in Y, and the enclosure of M in X contains Y, hence ca fails to be hereditary also in this case. The proof if based on the following easy

Claim. If X = X\ ]J^2 and for i = 1,2 M, is a non-empty set of Xi, then cx(Mi U M2) = cXl(Md U cx 2 (M 2 ). To finish the proof of (a) note that by the Claim c~1x(M) = l]_p Xv be the projection. Since £„ € c^J^Mi/) C C;f y (Mi,) for all

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v < a, one has

But if one had also x e c$(M) = (J,,
in contradiction to the choice of xv. (c) For MI C Xi and M2 C X2, first we show cXl (Mi ) x M2 C Cx (Mi x M2) ,

(2)

where .X" = X\ x X2. Indeed, for every a2 € M2 we may apply the continuity condition to the map X\ —> X, x\ H-> (x 1; a 2 ), to obtain CXI(MI) x {a2} C CX(MI x Af2) and therefore (2). Symmetrically to (2), one also has M\ x CX^MZ) C cx(M\ x M2). In this formula we can trade MI for CXI(MI) to obtain cXl(Mi) x c^2(M2) C cxCcx^Mj) x M2) C 4 (Mi x M2).

(3)

Since c^(^i) = (Jn^Cx^Mi) for t = 1,2, we get ^(Mi) x c X2 (M 2 ) = Un<.(c")x1(Mi) x (c")x2(M2). By (3) we get ^(Mi) x c X2 (M 2 ) C 4"(Mi x M2) C cx(M: x M2). This proves that ^(Mi) x c Xa (M 2 ) C c^-(Mi x M2), with the converse inclusion holding trivially. (d) follows immediately from (c). (e) is an easy induction proof. D Remark 3.1. (1) The assumption o(c) < oo in the concluding part of item (b) is essential. We shall show that c°° may be productive for a nonidempotent closure operator c with infinite o(c) (necessarily, o(c) = oo). (2) In item (c) one can replace c" by limit ordinal powers ca such that a is an indecomposable ordinal (i.e., (3 + (3 < a holds for every /3 < a). (3) Theorem 3.3 remains valid for c € CL(A) if the subcategory A is closed under the products and coproducts used in its proof. For every space X and M C X, let ax(M) be the set of points in X which are limits of sequences in M. This defines the sequential closure operator a of Top. It is productive and hereditary, but not idempotent. In fact, o(
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that for all a (2 < a i), aa is not hereditary, and that au and cr"1 are finitely productive, but not productive. In particular, cra'1 = <7°° gives an example of an idempotent operator which is not productive. Remark 3.2. For every X 6 A, let K,x be a set of subspaces of X such that X e K,x and, for all / : X -> Y in A and N € fCY, f ~ 1 ( N ) 6 f C x . Then

4(M) = f}{N e >CX : M C N} defines an idempotent closure operator c^ of A. This closure operator is hereditary if and only if for all M C Y C X 6 A with M 6 /Cy there exists L<= ICX with M = Lr\Y. If K*x ^ £x ftoZds /or every X £ A, then CK > cc. For any c € CL(A) one may take KX to be the set of c-closed subspaces of X , i. e., those M C X with cx(M] = M, and then obtain the corresponding idempotent closure operator CK . It is easy to see that c^ is the idempotent hull c°° of c in CL(A). Example 3.1. Each of the following properties for subspaces M C X defines a suitable set fCx as required in Remark 3. 2: (a) M closed in X (in the usual topological sense), (b) M open in X, (c) M clopen in X, (d) M a zero-set in X (so that M = f ~ 1 ( 0 ) for some continuous function f : X —> / into the unit interval), (e) M an Fa-set in X, (f) M = f ~ l ( N ) for some continuous function f : X —> R and some non-empty N C R. The corresponding closure operators CK are denoted by (a) k (the Kuratowski operator), (b) k* (the inverse Kuratowski operator), (c) q (see 1.2), (d) z (the zero operator), (e) p (the p-closure operator a), and (f) Q (the real closure13,) Since (b) 4= (c) =>• (d) =>• (a) => (e), one has k*z>k>p and z > Q > p. The inequality p < Q follows from the fact that every subset M of~K.is both Q-closed andp-closed, thus every Q-closed set of X is p-closed as well. Each of the six operators is idempotent and productive in Top, but only k and k* are hereditary. a Clearly px = kpx, where the topological space PX is the P-modiflcation of X. (Recall that a topological space X is called a F-space if every (7^-set of X is open.) b ln detail x & Qx(M) for X e Top, M C X and x £ X iff there exists a continuous function / 6 C(X) such that /(x) g /(M).

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Note that k* is conveniently described also by k*x(M) = {x£X: kx({x}) n M ^ 0}. When replacing k by an arbitrary closure operator c we can define more generally the inverse c* of c as c*x(M) = {x€X: cx({x}) n M ^ 0}.

4. Additivity and symmetry Definition 4.1. Let c be a closure operator of A, with (for simplicity) A as in 3.1. For a cardinal number K, c is called

• K-additive at X £ A if cx (\JieI M^ = \JieiCX(Mi) for all Mi C X, i € / card I < K.

c is • additive or a Cech closure operator if c is uj-additive; • fully additive if c is K-additive for every K. Note that K-additivity (with K > 1) entails cx(0) = 0. Only the trivial closure operator of Top does not satisfy the last condition (see 2.1). A fully additive operator is completely determined by its point closures, i. e. cx(M) = \J{cx ({x}) : x € M}. It is easy to see that on a space X with at most two points every closure operator is fully additive. An example of a three-point space with a non-additive closure operator is given below (Example 4.1 (2)). The property of K-additivity is stable under join and composition in CL(A). Consequently, every closure operator c has a K-additive core and a fully additive core, denoted by c+(-K^ and c®, respectively; we write c+ = c+(") These cores can be fairly easily computed as x

C

\jMiCX},

It is also easy to see that c+^ and c® inherit the properties of idempotency and hereditariness from c; on the other hand, /c-additivity is preserved when passing from c to the idempotent hull c°° (cf. Dikranjan-Tholen [6]).

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The inverse closure operator c* is always fully additive for every closure operator c £ CX(Top). Using this fact one can easily show that c*c* = c® and (cd)* = d*c* for any pair c, d € CI/(Top), and trivially, j* = j. Recall that a topological space X is an Alexandra i-additive in Top. p is the wi- additive core of k: p = k+^. (2) k A k* is an idempotent, hereditary and productive operator in Top which fails to be Cech. Hence the meet of Cech operators need not necessarily be Cech. To see this, consider the Alexandroff-Tucker topology of the set X = {0,1,2} with the usual order. Then the singleton {0} is k-dense and k* -closed, while the singleton {2} is k* -dense and k-closed. Therefore both singletons are k A k* -closed, while the set M — {0,2} is both k-dense and k*-dense. Consequently, I e kx(M) D k*x(M) = (k A k*)x(M], but (3) Since k A k* fails to be additive, its additive core b=(k/\k*}+ is properly finer than k A k*. This is the well-known b- or front closure, cf. Baron [2] (see also [12]); for M C X, bx(M) is the set of all points x with Ur\Mn{x} / 0 for every neighbourhood U of x in X . The closure operator b is, like the Kuratowski operator k, idempotent, hereditary, productive and Cech, but fails to be fully additive in Top. Consequently, the fully additive core b® is properly finer than b. One has b® = k® A k* = n.

Lemma 4.1. The real closure Q is u>i-additive with p < Q < z+"i. In particular, Q coincides with p for Tychonoff spaces.

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Proof. Assume X e Tych has countable pseudocharacter at some point x 6 X, i. e. {x} is a G^-set. Then one can find a continuous function / : X -> K such that {a;} = f ~ l ( { Q } ) . Then also X \ {x} = f~l(R\ {0}). We conclude that X \ {x} is g-closed. Therefore, if X has countable pseudocharacter then QX = Sx- In particular, Qx = s\ for a metrizable space X. To show that Q is wi-additive consider a family {Mn : n £ N} of ^-closed sets. Then for M = \Jn Mn and x e X \ (Jn g(M} there exist a sequence {fn : n € N} C C(X) such that fn(x) g fn(Mn). Consider the continuous product-map / : X —> EN. Since Y = R N is metrizable, gy is discrete, thus f ( x ) $ /(M) = \ J f ( M n ) , since /n(o;) 0 / n (M n ) (take the n-th projection !). By /(a;) £ /(M) = g(f(M)) we conclude that x £ g(M). To finish the proof note that Q < z is obvious, so that Q < z+u* follows from o>i-additivity of Q. D Remark 4.1. There exists a regular space X such that every continuous function f € C(X) is constant ([11]). Clearly for such a space X one has px < kx < Qx = 9x- Therefore, in the above corollary one cannot avoid the condition that X be a Tychonoff space. Actually, px < Qx fnoy hold even for a functionally Hausdorff space X. Indeed, the Tychonoff reflection rX of such a space X satisfies px = PrX and zx = zrx, while in general only px < PrX holds. Hence, if X is a space with px < PrX (e.g., Tychonoff's corkscrew), then also px < PrX = PrX = Px as rX is a Tychonoff space. Example 4.2. For a point x £ X denote by ax({x}) (resp. ex({x})) the arcwise connected (resp. connected) component of x in X. Now setting ax(M] = \Jx€Max(x) (resp. ex(M) = \Jx€Mex(x)) we get a closure operator a = {ax}x&ToP (e = {ex}x£ToP)- They both fail to be hereditary, but they are still weakly hereditary. (A closure operator is weakly hereditary ifcCx(M)(M) = cx(M) for all M C X e Top j It is easy to see that the closure operator a * k is not additive. This answers negatively a question from [6, p. 357, Table 1]: additivity is not preserved by cocomposition even when one of the closure operators is fully additive. A closure operator c e CL(A) is called symmetric if x e cx({y}) implies y € cx({x}) for all x,y e X e A. Symmetry is stable under taking meets and joins of families of closure operators. In particular, every closure

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operator has a symmetric hull, which is given by

Examples of symmetric closure operators of Top are a, q, p, z, b, //, while fc, cr, k* fail to be symmetric. For every fully additive closure operator c both compositions c*c and cc* are symmetric closure operators. Moreover, if c is also idempotent, then (csym)2 = c*c V cc*. Recall that a topological space X is symmetric if k\ is symmetric. It is easy to see that the symmetric Alexandroff- Tucker spaces are exactly the spaces X for which nx = 9x • Now we consider the symmetric hull w — ksym = k V k* of the Kuratowski closure k and its fully additive core w® = k® V k*. Lemma 4.2. Both w and w® are hereditary and (fully) additive, but fail to be finitely productive in Top (and are therefore not idempotent either, see Theorem 3.1 (2)). More precisely, o(w®) = w. Proof. Since k and fc* are hereditary, also w — k V k* is hereditary. Analogously, w® = k® V k* is hereditary. For M C X and x £ X, one has x £ w® (M) if and only if x £ {y} or y £ {a:} for some y £ M. In order to see that w is not finitely productive , we consider two copies of the (3-open set) Sierpinski space S = {0,1}, with {1} open, say. Then 0 € ws({l}) and 1 £ ws({0}), but (0,1) 0 w>sxs({(l,0)}). Since w® coincides with kV k* on finite spaces, this example shows that also w® fails to be finitely productive. Now Theorem 3.1 implies that neither w nor w® is idempotent. A direct proof of the latter property can be obtained by observing that for every power X = Sa one has w\wx = w<xwx = (^*)x = 9x, while wx 7^ 9x when a > 1. Indeed, consider the points 0, 1 £ X, where 0 (resp., 1) has all its coordinates 0 (resp., 1). Clearly, 0 is fc*-closed and fc-dense, while 1 is fc-closed and &*-dense. Now choose an arbitrary point x £ X distinct from 0 and 1. For v = 0, 1 let /„ = {i £ / : xt = t/}. Then both sets are non-empty by the choice of x and x = (x(°\ x^) £ 5/0 x S*1 , where all coordinates of x^ e S/0 are 0 and all coordinates of x^ € 5/J are 1. Then o;(0) £ Sr° is fc*-closed and fc-dense, while x(1) € Sfl is fc-closed and fc*-dense. Consequently, k"x(x) = {x(0)} x Sh and kx(x) = S!° x Consequently, w%(x) = ({o;(0)}x5/1)U(S'/0 x{z(1)}), in particular w^(x) 9 0, 1. This implies that w®(x) is w®-dense in X. Since the point x £ X

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was chosen arbitrarily, this proves that (tu®) 2 = gxTo prove o(w®) = LJ let us start by observing that w®w® = £VC, where £ = k®k* and £ = fc*fc®. An easy induction proof gives (iye)2n = £" V C™. Let Ln denote the partially ordered set Ln — {XQ,XI, ..., xn, yo, y\,..., yn} with XQ Xi < ... < 2/j_i > Xi < yi > xi+i < ... < j/n-i >xn
as the only non-trivial relations. Then XQ £ C,Ln(xn} Q (ty®)2" (xn) and y 0 _6 & n (y n ) C («;®)^(x n ), but x0 0 K)^- 2 (x n ) and y0 0 (tu®) 2 - / ™~ 2 (y n ). This proves that o(w®) > w. As for every fully additive closure operator, o(w®) < u. This proves the lemma. D Remark 4.2. (a) We shall see in Corollary 5.7 below that o(w) = oo. (b) It follows from Lemma 4-3 and item (c) of Theorem 3.1 that w = d*1 for no closure operator c. (c) The apparently fine closure operator (w®,) 2 turns out to be indiscrete on the powers of S. Since every T\ space is a subspace of such a power and since (if®) 2 is obviously discrete on every Ti-space X, this shows how strongly the square (u>®) 2 fails to be hereditary (while w® is hereditary). A similar comparison can be made for w: while w^, is indiscrete, w\ = k\ for every T\-space X. Now we find another property of w that provides a characterization of w:

Proposition 4.1. u; is the hereditary core of q in Top. Consequently, w is the largest proper hereditary closure operator of Top, and k is the largest proper hereditary closure operator o/Topi. Proof. Since w is hereditary, we get w < qhe- Now let c & CL(Top) be proper and hereditary. Then c < q (see 2.2). Consider x 6 cx(M) for M C X, and assume x £ k(M] U k*(M). Then M is a clopen set in the subspace Y = M U {x} of X, hence x $ M = qy(M). Since c < q is hereditary, this would imply x $ cx(M) - contradiction. Since k V fc*|ToPl = &|TC>PI, with Proposition 1.3, this proves also the second statement. D 5. Testing the Sierpiriski space Let c be a closure operator of A, in the setting of 2.1, and let X € A. We call c discrete (tame, wild, indiscrete) on X if ex = jx (ex = &x,

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cx = kx, ex = 9x)i i- e., if c is discrete (the usual Kuratowski closure, k*, indiscrete, respectively) on X. It turns out that the behavior of a closure operator on a particular space X may govern its behavior globally. The first space that we consider under this aspect is the Sierpiriski space S — {0, 1}, with 0, {!}, S open. Proposition 5.1. Let c £ CT(Top). Then : (1) c < k if and only if 03 < ks, and c < k* if and only if c$ < k^; (2) if c is Cech, then c c > fc® if and only if 03 = ks, and c = k* if and only if cs = k*s; (4) c > w® if ana ordy if cs = 9s ft- £-, c is indiscrete on S). Proof, (a) Assume that cs < ks, hence {0} is c-closed in S. Then for every X € Top and M C X the set F = kx(M) is closed in X, so that the characteristic function / : X -> S defined by /(F) = 0, f(X \ F) = I is continuous. Thus F = f~l(0) is c-closed in X, cf. 2.4. Hence cx(M) C kx(M) = F. The proof of the second statement is similar. (2) Follows from (1) since b = (k A k*)+. (3) Assume that eg = kg. According to (1) it suffices to show that c > k®. For X 6 Top and x 6 X we have to prove that kx(x) C cx(x). Let y £ kx(x), y =£ x and D = {x,y}. If D is T0, then D is homeomorphic to S and y 6 ko(x). Then our assumption gives y € CD(X) C cx(x). If D is indiscrete, then y 6 CD(X) as 03 = ks and there exists a continuous bijection S —* D. Now we conclude y e cx(x] as before. The proof for the case cs = kg is similar. (4) is left to the reader. D Theorem 5.1. Let c £ (7L(Top) be proper, hereditary and fully additive. Then : (a) (b) (c) (d)

c — k® if and only if c is tame on S. c = k* if and only if c is wild on S. c = p, if and only if c is discrete on S. c = k® V k* if and only if c is indiscrete on S.

Proof. To show cx (M) C c'x (M) for fully additive closure operators c, c' it suffices to consider singleton sets M, and when c, c' are also hereditary it even suffices to consider 2-point spaces X (since for arbitrary X one has

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x € cx({y}) iff x e cy({y}) for Y = {x,y} C X). Therefore, if we now wish to show that a given proper hereditary closure operator c coincides with one of w, k, k* or fj., then it suffices to show coincidence on 2-point spaces X since also these four closure operators are hereditary and fully additive. If X is discrete, then the coarsest proper operator q is discrete on X, hence all operators in question are discrete on X. If X is indiscrete, then the finest proper operator \i is indiscrete on X, hence all operators in question are indiscrete on X. Hence only the case X = S matters, and there are four possible cases : 03 = js = Us, Cs = ks = k f , GS = kg, and cs = gs = to® • a Corollary 5.1. The only proper hereditary and fully additive closure operators of Top are w®, k®, k* and ft. All but w are idempotent and productive. Corollary 5.2. The category Topo of TQ-spaces has exactly three proper hereditary and fully additive closure operators k®, k* and w, while the category Topi of TI -spaces has none. Proof. The proof of the above theorem shows ^|ibp0 = 3\Topa and k®\ToPl = k*\T0pl = W\TOPI = 3\Topi, so that these operators are no longer proper. In consideration of Proposition 2.3, this completes the proof. D

Our next objective is to obtain a further application of Theorem 5.1 in the case of closure operators on finite spaces. Note that there can be at most 2W closure operators since there are w finite spaces and each one can carry only a finite number of closure operators. Of course, the continuity condition in the definition of closure operator restricts severely this number, so that it would not be surprising if there were less than 2W closure operators in FinTop. Nevertheless, the following is proved in [9]: Theorem 5.2. [9] There are 2W distinct closure operators in FinTop. In Corollary 5.3 we carry out the classification in the case of hereditary and fully additive closure operators. By Proposition 2.4 the non-discrete closure operators in FinTop are determined uniquely by the closure operators in FinTop0 = Top0 (~l FinTop which is equivalent to the category of finite partially ordered sets with order preserving maps. These spaces are subspaces of Sn, n e N, where S is the Sierpiriski space {0,1} with open sets 0, {1} and S. According to 5.2

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there are 2W closure operators in FinTop, hence the restriction of closure operators to Topi, contrary to the case Topo, is strongly non injective (in fact, 2W closure operators which are non-discrete in FinTop will give the same restriction on Topi). Next we note that according to Corollary 5.3 the space S classifies also all hereditary Cech closure operators on FinTop since full additivity coincides with additivity on finite spaces. Corollary 5.3. The only hereditary proper Cech closure operators on FinTop (resp. FinTop0j, are k® V k*, k, k* and fj, (resp. k® V k*, k and k*). Next we give a partial result which does not make any additional assumption on the closure operator besides the necessary one, namely cs = k s. Lemma 5.1. If c € CL(Top) satisfies cs = ks, then cx = kx for every finite space X whose specialization partial order is total, i.e., S-tame yields X-tame for such spaces X. Proof. By Proposition 5.1 c\ < k\- Since we have a totally ordered set (X, <), for each pair x < y in X there is a monotone map / : S —» X with /(O) = x and /(I) = y. Since 0 e c,s({l}), this gives x € cx({y}). Therefore, cx > k\. D In order to characterize the Kuratowski closure operator in Top we need a property that describes a typical feature of direct products in Top. Let c be a closure operator of an epireflective subcategory A of Top. We say that c has the Tychonoff property (formerly, the finite structure property for products, abbreviated as f.s.p.p. [6]) if for every family {Xi : i £ /} in Top and for every M C X = fjj -Xi, a point x = (xi) e X is in cx(M) precisely when for every finite F C 7 one has KF(X) 6 CXF (TTJT(M)), with -KF '• X —> Xp = YlieF Xi the canonical projection. The Tychonoff property is stable under meet of closure operators. Therefore every closure operator c has a Tychonoff-hull cTych, namely a finest closure operator with the Tychonoff property that is coarser than c. It is easy to see that every finitely productive closure operator with the Tychonoff property is also productive. We have examples of closure operators without the Tychonoff property that are not finitely productive, such as w and w®(see 4.2), but no answer to:

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Open Problem 1. Does there exist a closure operator of Top with the Tychonoff property which fails to be finitely productive ? The use of the Tychonoff property becomes clear by the following theorem which has been proved in [6] in a more general context. Again for the reader's convenience we give a proof in the context of this paper. Theorem 5.3. Let d € CL(A) have the Tychonoff idempotent closure operator d> c is productive.

property Then every

Proof. In order to check productivity of c fix a product X = lite/ -^t and a subspace M = Yliei Mi, where Mj C Xi for each i € /. For J C /, let Xj = Hie j Xi and Mj = Yli€J Mi and write cj(Mj) instead of cxj (Mj) (J £ /), and in particular Ci(Mj) instead of cXi(-Wt) (« € 7). Now d < c implies c < cd < cc, hence c — dc since c is idempotent. Therefore, in order to show x € cx(M) for x £ Hie/ ci(Mi) , it suffices to show x e dx(cx(M)). But since d has the Tychonoff property , for that we need to show only 717? (z) 6 dp(-KF(cx(M))) for every finite F C /. In fact, since M = MF x M/\F and since c is finitely productive (see Theorem 3.3(c)), one has

cx(M) = cF(MF) x and therefore 7r^(cx(M)) = CF^F). Furthermore, applying finite productivity again we conclude that CF(MF) = Hi£Fci(Mi)- Therefore, x e YlisICi(Mi) implies nF(x) 6 CF(MF) = TTF(CX(M)) C

dF(^F(cx(M)}),

as required.

D

Obviously k has the Tychonoff property while the discrete closure operator does not have it. Corollary 5.4. Every idempotent closure operator c> k is productive. Idempotency is essential in this corollary (e.g., for the ^-closure, as defined after Theorem 5.4 below, 0" is not productive, see Theorem 3.1). Corollary 5.5. Let c & CL(Top) have the Tychonoff is productive.

property. Then c°°

The above corollary shows again that the sequential closure a does not have the Tychonoff property.

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Corollary 5.6. // c G CL(Top) is a (non-idempotent) closure operator with (jj < o(c) < oo, then there exists no closure operator d < c with the Tychonoff property. In particular, k j£ c, and no power of c can have the Tychonoff property. Proof. Arguing for a contradiction assume that there exists a closure operator d < c with the Tychonoff property. Then c°° is productive by Theorem 5.3. On the other hand, Theorem 3.1 implies that c°° cannot be productive as u < o(c) < oo, a contradiction. D

Corollary 5.7. Let a closure operator c> k of Top have o(c) > uj. Then o(c) = co. In particular, o(w) = 0(6) = oo. Proof. Follows directly from Corollary 5.6.

D

This corollary applies also to a proper class of closure operators 5(77) > k of infinite order, depending on a total order 77, as defined in [6, 8]. Open Problem 2. Does there exist a closure operator d of Top with the Tychonoff property and d < k? Corollary 5.5 shows that the Tychonoff property, along with idempotency, implies productivity. But the Tychonoff property is surely not a necessary condition for productivity: a does not have it since otherwise its idempotent hull cra<1 would have to be productive, but it is not (Theorem 3.1). In what follows we give explicit examples which show failure of the Tychonoff property not only for Xi x . . . xXn (n € N) and consider yk € X defined by yk(i) = i for 1 < i < k and yk(i) = k for k < i < oo. For M = {yk : k € N} and x £ X defined by x(n) = n for each n e N, one has ?rn(x) = 7rn(2/n) £ irn(M}. Hence Kn(x) € 7T n (M) = j(7T n (M))

(4)

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for each n € N. On the other hand, x g M and X is Hausdorff. This proves that (5)

for every closure operator c such that ex = jx, e.g., j, /u, b, k* A k, w®, k® or k*. By (4) and (5) none of these closure operators has the Tychonoff property. To prove that w does not have the Tychonoff property we need a different example. Let T — {0,1,2} be equipped with the AlexandroffTucker topology of its usual order. For every n e N let Xn = T, T " = Unxn, Tn = Hl=1Xk and wn : T" -> Tn be the canonical projection. Define x = (xn) £ Tw by letting xn = 1 for every n e N and 3,(fc) = (ylfc)) e T" by letting y(n} = 2 if n < k, otherwise y£} = 0. Finally, let M = {j/(fc> : k 6 N}. Then •nn(x) e fcJn(7rn({7/(")}) C fcJn(7rn(M)) C w^(7r n (M)) for all n e N. On the other hand, x $ WT"(M). Indeed, since W = {0,1} x Yl%!,2Xk is an open neighbourhood of x missing M, we have x 0 kT*(M). On the other hand, for every n € N the set U = Tn x {0} x fl£Ln+2 ^k is an open neighbourhood of j/n^ in Tu missing the point x, thus j/ n ) ^ fcr^({a;}). This proves that x ^ fcyu,(M), and consequently x g wr^(M). This example cannot be used for a since x £
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Remark 5.1. (a) The first example of 5.1 does not work for p, since x 6 px(M). It follows from Theorem 5.4 below that p does not have the Tychonoff property since it is hereditary. But the reader should note that p has the countable Tychonoff property defined in analogy with the Tychonoff property, but w.r.t. countable subproducts. (b) The second example of 5.1 shows that every closure operator c € CL(Top) with k* SF is the canonical projection. Since SF is finite, c is fully additive on SF. Hence, according to Theorem 5.1 we have several cases depending on whether c is discrete, indiscrete, wild or tame on SF. By Theorem 5.1 these correspond to the closure operators fj,,w,k* and k®. Since neither of k*,w,w® and \i has the Tychonoff property by Example 5.1, we are left only with the possibility that c coincides with k on all finite powers SF. Therefore, since both c and k have the Tychonoff property, but k® does not, c and k must coincide on X as well. n Of the four characteristic properties in Theorem 5.4, certainly neither properness nor the Tychonoff property can be dropped (consider g and a; note that both closure operators are productive, Cech, and hereditary, but a does not have the Tychonoff property according to 5.1). But also hereditariness is essential : Velichko's ^-closure ([14]), defined by 9x(M) = {x € X : U(~\M / 0 for every closed neighbourhood U of x in X}, is a proper, productive Cech closure operator with the Tychonoff property, which is neither hereditary nor idempotent. We do not have an example showing the essentiality of finite additivity for 5.4.

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6. Testing the reals Now we define further closure operators of Top which will be essential in this section. Each of them depends on a triple (P,Q,v) consisting of a "pattern space" P, a subspace Q C P and a point v € kp(Q). For a subset M of a space X and x e X let x & c^Q'v(M) when there exists a continuous function / : P —> X such that f ( v ) — x and f ( Q ) C M. It is easy to see that cp>Q'v is a productive closure operator of Top that in general fails to be additive, idempotent or hereditary. Several particular instances are of major interest. (i) One obtains the sequential closure operator as a = cpx(M) for M C X if there exists a continuous function / : [0, 1] -> X with /(O) = x and /(1/n) 6 M for each n 6 N). Then a is additive, but not hereditary. (iii) We abbreviate c[°,i],(o,i],o by A ( L e - ) x e \X(M) for M C X if there exists a continuous function / : [0, 1] —> X with /(O) = x and /((0, 1]) C M). Then A is hereditary but non-additive. (iv) The closure operator 5 = C^'^'Q'° with Q = (0, 1] n Q satisfies S < a * cr. It is neither hereditary nor additive. (v) More generally, let C C (0, 1] be a subset such that 0 6 fcp(C) and 0 e MQ), where Q '•= (M] \ C. Set A( c > := c^W'0 with the following relevant example A^ := A^ with C = {1/n : n 6 N}. This closure operator is neither hereditary nor additive. Obviously, p, < k* < A < A^) <5
(6)

The following easy example shows that in general (6) is not an equality even for metrizable separable spaces X. Example 6.1. Actually, one can refine (6) as follows: a
IV

J

of the plane and M = {(1, 0)} U (X \ [0, 1] x {0}). It suffices to check that M is a* a-closed, while (0,0) € <7x(M) A a

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Proposition 6.1. Let X be a locally arcwise connected space. Then ax = qx, so that (6) becomes an equality for X if and only if ax — GX- This occurs when X is also first countable. Proof. We note that now each component ax(x) is a clopen set, thus coincides with qx(x). Then arbitrary unions of quasi-components qx(x) will be clopen as well. Hence kx < ax = qx- Now equality in (2) is equivalent to ax = o~x since always cr\ < qx • Assume that X is first countable. Let x e o~x(M) A ax(M) for some subset M of X. Then this must be witnessed by a sequence xn —> x with xn e M and by some m € M such that x e ax({m}}. As the latter set is clopen, we can assume without loss of generality that all xn e ax({m}) and consequently xn £ ax({x}). Since x = \imnxn, we can choose a subsequence y^ = xnit and a decreasing sequence of neighbourhoods U\ D U% 3 ... D Un 2 ... of x forming a base of the neighbourhood filter of x and such that there exists a continuous function fk '• [0,1] —> X with image contained in Uk such that /fc(0) = x and //t(l) = J/fc for every k € N. Making use of the functions fk one can define a single continuous function / : [0,1] -> X with /(O) = x and /(1/fc) = yfc 6 M for each k & N. This proves that x £ ax(M). The equality (5) is proved. D In the following lemma we compute the hereditary hulls of these closure operators. Lemma 6.1. ahe = a, ahe = ehe = qhe = g hold both in Top and Tych. Proof. Easy extension spaces (which remain in the respective subcategory) provide the necessary inclusions. D Let us recall that a closure operator c is R-discrete (resp. R-tame, Rindiscrete) if CR = JR (resp. if CR = &R, resp. if CR = &R iff c> a. (iv) CR = JR (i.e., c is R-discrete) iff c< g.

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ET AL.

Proof. Let X be an arbitrary topological space. (i) Assume that c is R-indiscrete and let x £ ax({y})- Then there exists a continuous function / : [0, 1] —> X such that /(O) = x and /(I) = y. One can easily extend / to a continuous function / : R —> X. Then 0 e CR(!) yields x = /(O) e cx({y}), hence ax({j/}) C cx({y}) = c%({y}). This proves that ax < c®. Since X is arbitrary this implies a < c. Clearly, a R such that g(A) = {0} and g(x) = {!}. Since CR is not indiscrete, 3(0;) = 1 ^ c({0}) = c(<7(A)). Since g is "c-continuous" this yields x £ cx(A). The inverse implication is again trivial since ZM = fcm. (iii) Let a; 6 a\(M) be witnessed by the continuous function / : [0, 1] —> X with /(O) = x and /(4) C M, where ^ = {1/n : n e N}. Let /i : R -> R be defined by h(y) = y for all y € [0, 1], h(y) = 1 for y > 1 and /i(y) = 0 for y < 0. Then g = f o h : [0,1] -* X satisfies g(A) C M. Moreover, 0 € kyt(A) C CR(A) by hypothesis, so x = 5(0) £ S(PR(A)) C c x (s(A)) C c x (M). (iv) Let X be a Tychonoff space and assume that M C X is ^-closed. We have to show that M is also c-closed. Since every ^-closed set of X is an intersection of inverse images of subsets of R under continuous maps, it suffices to observe that subsets of R are c-closed by hypothesis. This proves ex < Qx • The inverse implication follows trivially from the equality

Remark 6.1. (1) Obviously, one cannot improve item (iii) by replacing a by a coarser closure operator d (like a f\ a, for example), since (iii) applied to c = a (this is possible as a^ = k^) would give a > d. (2) By (i) and Proposition 6.1, c < g and c is ^-indiscrete if and only if a < c
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that an R-indiscrete closure operator c satisfies 03 = gs (compare with the previous section). (b) R-tame closure operators: 6, k, K, a, a. More generally, any closure operator c satisfying a < c < z is R-tame. (c) R-discrete closure operators: g, p, k*, b, \JL. More, generally, any closure operator finer than p or k* is R-discrete. We will separately classify the R-discrete, the R-tame and the Rindiscrete closure operators on spaces close to being metrizable. Let us first see that these are the sole possibilities for a Cech closure operator of Top: Theorem 6.1. Let c be a closure operator of Top. Then: (i) Either c is R-indiscrete, or CR < MR(ii) Assume c is neither R-indiscrete nor R-tame. Then CR is Cech if and only if c is R-discrete. Proof, (i) Follows from item (ii) of Lemma 6.2 with X = R. (ii) Under this assumption on the closure operator c there exists a converging sequence an —> a in R such that (6)

where A = {an : n € N}. It suffices to prove that for each point x € R we have x $ CR(R \ {x}). To this end we "split" R \ {x} into a union R \ {x} = M U N and we prove that x £ c^(M) and x £ CR(./V). Then additivity of CR applies. For n s N set Fn = {y 6 R : l/(n + 1) < min{|o; - y\, 1} < l/n}. Now define / : R -> R by f(y) = min{|x - y|,l}. This is a Lipschitz function, so in particular (uniformly) continuous. Its range is in fact [0, 1]. Next we compose / with the piece-wise linear continuous function h : R —> [0,1], such that ft((-co,0] = a, /i([l/(2n + l),l/(2n)] = an (n 6 N) and h([l, +00) = ai. Then the function h will be continuous. The composition g = hf : R —> R sends the set M = (JneN ^2n into the set A. Since g(x) = a and /(M) C A, (6) yields (by the continuity property of closure operators, cf. Definition 2.1) that x £ cm(M). Analogously one proves that x $ c®.(N), where N = IJneN Fin-\- Since obviously R \ {x} = M U TV, we are done.D Remark 6.2. One can show that there exist infinitely many non-additive closure operators that are distinct on R. More precisely, for c €. CX(Top)

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CK is not Cech if and only Z/CR = AR or AR < CR < JR. Appropriate choices of C give c-many closure operators of the form AR ' that are distinct on K. (see [7]). 7. Characterization of the Kuratowski closure in smaller subcategories Here we characterize the Kuratowski closure as a Cech operator in smaller subcategories of Top. First we classify Tychonoff spaces of countable pseudocharacter by means of R-discrete closure operators. Theorem 7.1. For a Tychonoff space X the following are equivalent: (i) every R-discrete closure operator of Top is discrete on X, i. e., R- discrete yields X-discrete; (ii) X has countable pseudocharacter. Proof. Apply Lemma 6.2.

a

Now we give the same result from the point of view of the closure operator. Corollary 7.1. A closure operator c on Tych is ^-discrete if and only if it satisfies c < p. In particular, c is discrete on every Tychonoff space of countable pseudocharacter. A space X is submetrizable if X admits a coarser metrizable topology. Corollary 7.2. Let c be an ^.-discrete closure operator of Tych. Then it is X-discrete for every Tychonoff space which is either first countable or submetrizable. One can be rather satisfied with this classification of JR-discrete closure operators since p is the u>i-additive core of fc, so that we have again a kind of characterization of k. Now we classify R-indiscrete closure operators. Corollary 7.3. Let X be a locally arcwise connected space and c be a proper ^.-indiscrete closure operator of Top. Then kx < ax = ex = 1x • Consequently, GX = kx iff X is a symmetric Alexandraff-Tucker space.

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Proof. By Proposition 6.1 ax = qx- Now Lemma 6.2 applies to get kx < o-x = ex = Qx- The conclusion follows from Proposition 4.1 since the hypothesis implies that k\ is fully additive and symmetric (as ex =
D

It would be interesting to characterize the class A (B) of Tychonoff spaces (Frechet-Urysohn Tychonoff spaces, respectively) X such that ax — ax • Since ax < ex. < kx, for Tychonoff spaces, then obviously ax < ex < kx will hold for each space X € A and any R-tame closure operator c, and so ex = kx for spaces X e B. The next theorem characterizes k as the only proper, hereditary, Cech closure operator on Prechet-Urysohn spaces. Theorem 7.3. Let c be an E.-Cech closure operator of Top, such that the restriction of c on the subcategory of Tychonoff spaces of countable pseudocharacter is proper. Then c FU = &|FU °, iff the restriction C|FU is hereditary Proof. According to Theorem 6.1 there are three possibilities for c. Our first aim is to rule out two of them. Assume that c is R-indiscrete, i. e. CR =
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One may ask how strong is the condition of hereditariness. The following example shows that the conclusion of the above theorem, i. e. c\ = kx, does not remain valid even for separable metric spaces if the closure operator is not hereditary. Example 7.1. Let X be the disjoint sum of an interval and a converging sequence. Then one has jx < ctx < kx • Theorem 7.4. Let c be a Cech closure operator of Top. Ifcx is proper on X, then GX coincides with the Kuratowski closure operator kx for every first countable Tychonoff space X which is either connected and locally arcwise connected or zero-dimensional. Proof. In case X is connected and locally arcwise connected, c\ = 1x by Corollary 7.3 when c is R-indiscrete. Since X is connected, qx — 9x, a contradiction. If c were R-discrete, then Cx = jx by Corollary 7.1. Therefore c must be R-tame. Then by Theorem 7.2 ax < ex < kx- Since (xx — a in X such that a#cx(A),

(7)

where A = {an : n € N}. Now we are ready to prove that under this assumption the closure operator ex is discrete. It suffices to prove that for each point x e X we have x £ cx(X \ { x } ) . To this end we "split" X \ {x} into a union X \ {x} = M U N and we prove that x £ cx(M) and x $ cx(N). Then additivity of c applies. For every n e N choose a clopen neighbourhood Fn of x such that the family {Fn : n e N} is a proper descending chain and forms a base of the neighbourhood system of x. Set M = U n6N Fin \ Fzn+i and M' = M U {x}. Now define / : X -> X by f(x) = a = f[X \ M'], f[F2n \ F2n+1] = an for n e N. This is a continuous function which sends M into the set A. Since f(x) = a and /(M) C A, (7) yields (by continuity of closure operators, cf. Definition 2.1) that x $ cx(M). Analogously one proves that x $ cx(N), where N = UneN Fzn-i \ F^n- Since obviously X \ {x} = M U TV, we are done. D

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Remark 7.1. (a) By Example 7.1 one cannot omit local arcwise connectedness (zero-dimensionality, respectively) of the metrizable space X to claim ex = kx- On the other hand, one cannot omit first countability of the space X even when X is a connected and locally arcwise connected topological group, while the closure operator is even hereditary (take X = (R/Q) Wl , then ax < kx). (b) To see that additivity is essential in the above theorem it suffices to take c = cp'Q'v, where P = [0,1) R Q, v = 0 and Q = P \ {v}. This is a hereditary non-additive closure operator strictly finer than k. Corollary 7.4. All proper Cech closure operators of Top coincide on a connected locally Hilbert space (= Q-manifold) with the Kuratowski closure. Corollary 7.5. Let X be one of the spaces R" or £3. If a Cech closure operator c of Top is proper on X, then ex coincides with kxRoughly speaking, we have shown that a closure operator on a locally "good" space X must coincide with the Kuratowski closure. References [1] A. V. Arhangel'skii and S. Franklin, Ordinal invariants for topological spaces, Mich. Math. J. 15, (1968) 313-320. [2] S. Baron, Note on epi in T0, Canad. Math. Bull., 11 (1968) 503-504. [3] D. Dikranjan, Semiregular closure operators and epimorphisms in topological categories, Suppl. Rend. Circ. Mat. Palermo, Serie II, n. 29 (1992) 105-160. [4] D. Dikranjan and E. Giuli, Closure operators I, Topology Appl., 27 (1987) 129-143. [5] D. Dikranjan, E. Giuli and W. Tholen, Closure operators II, in: J. Adamek and S. McLane ed., Proceedings of the International Conference on Categorical Topology, Prague, 1988 (World Scientific, Singapore 1989) 297-335. [6] D. Dikranjan and Walter Tholen, Categorical Structure of Closure Operators with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and its Applications, vol. 346, Kluwer Academic Publishers, DordrechtBoston-London 1995, pp. 358+xviii. [7] D. Dikranjan, Walter Tholen and Stephen Watson, Classification of closure operators for categories of topological spaces, II: non-additive closure operators, regular closure operators and weak hereditariness, work in preparation. [8] D. Dikranjan and Stephen Watson, The Category of S (a)-Spaces is not Cowellpowered, Topology Appl., 61 (1995) 137-150. [9] D. Dikranjan and Stephen Watson, Finite topological spaces and their closure operators, work in progress. [10] R. Engelking, General Topology, Revised and Completed Edition (Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989).

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[11] H. Herrlich, Wann sind alle stetigen Abbildungen in Y konstant ? Math. Zeitschr. 90 (1965), 152-154. [12] L. Nel and R. G. Wilson, Epireflections in the category of TQ-spaces, Fund. Math. 75 (1972) 69-74. [13] S. Salbany, Reflective subcategories and closure operators, Categorical topology (Proc. Conf., Mannheim, 1975), pp. 548-565. Lecture Notes in Math., Vol. 540, Springer, Berlin, 1976. [14] H. Velichko, H-closed topological spaces, Mat. Sb.(N.S.) 70(112), (1966) 98-112 (Amer. Math. Soc. Transl. 78, Ser. 2 (1969) 103-118.)

TOPOLOGICAL STRUCTURES IN LOGICS

WERNER GAHLER Institut fur Mathematik I Freie Universitdt Berlin Arnimallee 3 14195 Berlin, Germany E-mail:gaehler@rz. uni-potsdam. de In this paper the classical notion of modal logic is extended to a more general type related to monoidal predicate logics. Mathematics Subject Classifications (2000): 03B45,03B60,06D20, 06D30,06D35,54A05 Keywords: Classical modal logic,monoidal logic,monoidal predicate logics,generalized topological structures.generalized modal logics

1. Basic structures of monoidal predicate logics In the following we list some important types of these basic structures. 1.1. Residuated ilc-monoids The most general type of basic structure is that of a residuated ilc-monoid, which is a quintuple (X, 1, <-^, *--", •) consisting of a lattice (X, 1, ^-, *--) with maximal element 1, which moreover is equipped with a commutative and associative binary operation • : X x X —> X, such that the following holds: 1) (X, 1,'—• , •—', •) is integral, that is, 1 • a = a holds for all a 6 X. 2) (X, 1,^, •—",•) is lattice ordered, that is, a b = max{ c£Xa-c
exists.

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ilc means that the monoid is integral and lattice ordered and the monoid operation • is commutative. In general it is open wether the smallest element 0 exists or does not. If it exists, then for each a € X we may define -a = a -> 0,

(2)

called the negation of a. Frequently it will be assumed that a residuated ilc-monoid is nondegenerate which means that 0 exists and differs from 1. Applying residuated ilc-monoids in non-classical predicate logics there always will be assumed that they are complete where of course completeness is meant with respect to the lattice operations —• and ^.

1.2. Non-empty Hey ting-algebras By a non-empty Heyting-algebra we mean a residuated ilc-monoid (X, 1, ^, •—', •) for which • = ^ .

1.3. Frames Non-empty complete Heyting-algebras will also be denoted as frames.

1.4. Girard-monoids A Girard-monoid is a residuated ilc-monoid such that the smallest element 0 of X exists and the rule of double negation holds, that is a = a

(3)

holds for all a € X. 1.5. Commutative quantales A commutative quantale (X, H , \J , •) is a non-degenerate complete lattice equipped with a commutative and associative binary operation • such that for each element a of X and each family (bi)ie/ of elements of X we have bi)-

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1.6.

101

MV-algebras

An MV-algebra is a Girard-monoid which fulfills the divisibility condition, which means, that for all a, b € X for which a < b holds, there is an element c of X such that

a — c-b .

(5)

1.7. Boolean algebras A Boolean algebra is an MV-algebra for which • coincides with ^ . 2. Associated monoidal predicate logics In the classical predicate logic the logical signs are the following: • the prepositional connections -i, A, V, —> and <-> • the quantifiers V and 3 • the equality symbol = In the non-classical logics moreover a binary junctor ® is added. ® is related to the binary monoid operation • . In the following we list some important types of monoidal predicate logics and add to each of them the related basic structure: • • • • •

Monoidal predicate logic - residuated ilc-monoids Intuitionistic logic - non-degenerate Heyting-algebras Linear logic according to Girard - Girard-monoids Lukasiewicz-logic - MV-algebras Classical logic - Boolean algebras

3. Relations between the basic structures

Each arrow in Figure 1 goes from some type of basic structures of monoidal predicate logics to another more general type. The bold-faced arrows only connect complete basic structures with complete basic structures. Proposition 1. For any two arrows in Figure 1 with the same starting point the following holds: The basic structure at this point is exactly the basic structure given by combinig both the basic structures at the related arrow heads.

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GAHLER residuated ilc-monoids

Girard-monoids non-empty Heyting-algebras

MV-algebras

commutative quantal(

compl. Gir.-mon

X

frames

compl. MV-, Boolean algebras

complete Bool. alg. Figure 1. The diagram of basic structures.

4. t-Norms

The notion of t-norm has been introduced by B. Schweizer and A. Sklar (see Ref. [5]). A mapping T : [0, 1]2 —» [0, 1] is called a t-norm if for all a, 6, c, d € [0, 1] we have T(o,l) = a T(a,b) = T(b,a)

a T(a, c) < T(b, d) 4.1. The related commutative quantales Proposition 2. For each t-Norm T which is continuous, ([0, 1], 1, <,T) is a commutative quantale, in particular bi) =

\jT(a,bt)

TOPOLOGICAL STRUCTURES

103

holds for each element a of [0,1] and each family (fei)jg/ of elements of [0,1].

4.2. Some standard t-norms A lot of different classes of t-norms are well-known. As standard types of t-norms we mention here the following: • • • • •

the the the the the

standard-fuzzy intersection Tm;n, algebraic quotient Tquo, algebraic product Taig, bounded difference (due to Lukasiewicz) Ttuk and drastic intersection Tdra-

They are defined for all a, b € [0,1] as follows: T m in(a,b) = min{a,6} , f f fc i f a f e ^ O T quo (a,6) = < a + b-ab (0 otherwise, Taig(a, 6) = ab, T Lu k(a, 6) = max{0, a + b - 1} and

{

a if 6= 1, b if o = l, 0 otherwise.

Of course, Tquo is well defined: for all a, b e [0,1] we have a + b - ab = 0 if and only if a = b — 0. With except of Tdra the mentioned standard t-norms are continuous mappings. 5. Interior operators of basic structures

Let X = (X, 1, ^, *—•, •) be a residuated ilc-monoid. An interior operator on X is a mapping int: X —> X such that (PI) (P2) (P3)

int(a) < a holds for all a £ X. int(l) = 1. int(a • 6) = int(a) • int(6) holds for all a, b € X.

GAHLER

104

An interior operator int on X is also called a pretopology on X. If int o int = int

holds, then int is called a topology on X. If 0 exists, then each interior operator int can be associated the closure operator cl = - int - .

(6)

If the rule of double negation holds, the closure operator is associated in a one-to-one way to the interior operator. In this case we have int = — cl —

6. Examples of interior operators and closure operators defined by t-norms

6.1. The case of standard-fuzzy

intersection

We have that ([0,1], <, Tm;n) is a frame. Example 1. An example of int: [0,1] —> [0,1] for case 6.1 is the following: Fix any real number c £ [0,1] and for each a € [0,1] let min{a, c} if a < 1

int (a) =

1

if a = 1 .

(7)

int

0

c

1

Figure 2. The interior operator of example 6.1

In this example we have min(min(a, c),c] = min(a, c)

(8)

TOPOLOGICAL STRUCTURES

105

if a < 1, hence int is a topology: int o int = int. Since we have 1 if a = 0

-a =

(9)

0 otherwise, it follows 1 ifa = 0 a -—- — a =

a otherwise. Thus we have a ^ — a = 1 only for a = 0 and a = 1,

(10)

therefore also int(o •—' —a) = 1 only for a = 0 and o = 1. (6), (7) and (9) implies ' 0 if a = 0

cl(a) =

(11)

1 otherwise. Hence cl does not define int.

cl

0

1

Figure 3. The closure operator in the example of 6.1

(9) and (11) implies 0 if a = 0

-cl(-o) =

1 otherwise.

(12)

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GAHLER

6.2. The case of algebraic quotient We have that ([0,1], <,T quo ) is a commutative quantale. Example 2. An example of int: [0,1] —> [0,1] for case 6.2 is the following: For each a e [0,1] let int(a) =

(13)

2-a

Since for each a 6 [0,1] we have

0

1

Figure 4. The interior operator in the example of 6.2

(into int) (a) =

4-3a '

(14)

in this example int fails to be a topology. Because of —a =

' 1 if a = 0

0 otherwise,

(15)

it follows

cl(o) =

f 0 if a = 0

I 1 otherwise.

(16)

Hence, the closure operator cl is defined as in the example of 6.1 (see Figure 3) and as in this example, cl does not define int. As in the example of 6.1 only for a = 0 and a — 1 we have a -—- —a = 1 and hence int(o •—- —a) = 1.

TOPOLOGICAL STRUCTURES

107

6.3. The case of algebraic product We have that ([0,1], <,T a i g ) is a commutative quantale. Example 3. An example of int : [0,1] —> [0,1] for case 6.3 is the following : Let K be a real number > 1. For each a € [0,1] let int(a) = aK .

(17)

In this example int is not a topology. For each a & [0,1] we namely have

0

1

Figure 5. The interior operator in the example of 6.3

(into int) (a) = aK'K

(18)

Since —a =

1 if a = 0

0 otherwise,

(19)

it follows

cl(a) =

' 0 if a = 0 (20)

1 otherwise. Hence, even in this case the closure operator cl is the same as in both the examples of 6.1 and 6.2 and we also have that cl does not define int. Moreover as in these examples only for a = 0 and for a = 1 we have int(a •—• -a) = 1.

GAHLER

108

6.4. The case of bounded difference We have that ([0,1], <, TL U R) is a complete MV-algebra. Example 4. An example of int : [0,1] —> [0,1] in the case of 6.4 is the following: Let int(a)=0 if 0 < a < -, &

1 1 3 int(a) = - if - < a < -, 3 3 7 int(a) = - if - < a < -, 7 7 1 5 int(a) = - if - < a < — o

o

ID

and so on. In general, for n = 0,1,... and a G [0,1] with

2 n+1

< a <

we have int (a) =

(21)

2™

Moreover, let int(l) = 1.

(22)

Prom (21) and (22) it follows that in this case int is a topology: intoint =

int

3 4

7..1 8 -1-

Figure 6. The interior operator in the example of 6.4

int.

109

TOPOLOGICAL STRUCTURES

For the negation of each a e [0,1] we have -a = 1-a,

(23)

moreover we have cl(o) = l-int(l-a),

(24)

int(o) = l - c l ( l - a ) .

(25)

hence also Even in this example only for a = 0 and for a = I we have a *—• —a = 1

cl

i8 i4

i2

Figure 7. The closure operator in the example of 6.4

and hence int(a -—- —a) = 1.

6.5. Summarizing of some properties of the examples in 6.1 to 6.4 Table 1. The properties stated in this table follow for T m j n by means of (8), (9), (11), (12), for T quo by means of (14), (15), (16), for Taig by means of (18), (19), (20) and for T Luk by means of (21), (22), (23), (24). Ta

rule of double negation cl defines int int is a topology type of residuated Uc-monoid

Tl.uk

no

no

no

yes

no

no

no

yes

yes

no

no

yes

frame

commut. quantale

commut. quantale

complete MV-alg.

1

110

GAHLER

7. On modal logics 7.1. The classical case of modal logic A basic structure of the classical modal logic is a pretopological complete Boolean algebra, that is, it is a complete Boolean algebra X = (X, 0,1, — , H'U) equipped with an interior operator int : X —> X. |"| and (J are the symbols used for describing the infima and suprema of the subsets of X. int is denned as in chapter 5 in the more general case of a residuated ilc-monoid. As in this more general case int is also called a pretopology and it is called a topology provided that int o int = int. Clearly, — means the negation as denned in (2), hence for each a € X, —a denotes the negation (J c of a. a--~c = 0

In the classical modal logic additionally to the logical signs -i, A, V, —>, <->, V, 3 and = of the classical predicate logic a further logical sign is to add: namely a unary junctor denoted by I. For each formula A in this type of logic the symbol IA is to read as follows: "it is necessary that A". Note that in case of the classical modal logic the binary junctor cg> is not necessary to be presented separately since in this case it coincides with the logical sign A. Using the prepositional connection -i (see section (2)) for each formula A the formula -i\-iA will also be denoted by CA The symbol CA is to read as follows: "it is possible that A". We obtain an axiom system of the classical modal logic in adding to any axiom system of the classical predicate logic the following axioms (Ml), (M2), (M3) and (I), in which A and B are arbitrary formulas: (Ml) \A->A, (M2) (AV->A)-* \(AV- I ( A A B ) , moreover the inference rule (I) A±J1 L4->IB 1;

Axiom (Ml) can be read: If A necessarily holds, then A holds. Of

TOPOLOGICAL STRUCTURES

111

course we also have: If A holds, then it is possible that A holds. (Ml) implies that each interior operator int has the property: inta < a. (M2) together with (M3) and (I) implies that each interior operator int has the property: intl = 1. Because of (M3) together with (I) we have that each interior operator int has the property: inta -—• mtb = int(a ^ 6).

7.2. The general case of modal logic A basic structure in the general case of modal logic is a pretopological complete residuated ilc-monoid, that is, it is a complete residuated ilc- monoid X = (X,l,f\ ,\J , •), equipped with an interior operator int : X —» X . Because of the completeness the smallest element 0 always exists. We assume that 0 / 1 holds. In the general case additionally to the unary junctors I and C as a further logical sign the binary junctor appears, which is related to the binary monoid operation • . In the following we present at first an axiom system of the monoidal predicate logic, where A, B and C are arbitrary formulas: (Tl) (A^B) -* ((B -» C) -» (A -> C)) (syllogism). (T2) A-> (AVB). (T3) B-> ( A V B ) .

(T4) (T5) (T5') (T6) (T6') (T6") (T7)

(A-+C) -> ((B^C) - ((4 VB) ->A (A&B) ->A (A AS) ->B. (A®B) -> (B®A). (A®(B®C)) -> ((A®B)®C). (C-+A) -» ((C-^B) -> ( C - ( A A B ) ) ) .

(T8) (>4 -» (B -» C)) -» ((yl®B) -> C) (importation). (T9) ((4®B) -> C) -» ( ^ - » ( B - * C 7 ) ) (exportation).

(T10) (A®-iA) -* J5 (Duns Scotus). (Til) (4-> (A »-->!)) -* -,>!. (MP) --—-- (modus ponens). U

—

m case x

^oes n°t appear free in A, (first quantor

rule). ^_ _> (Q2) -—---— in case x does not appear free in B, (second quantor 3X./T. —

rule).

&

112

GAHLER

Clearly changing here ® to A gives an axiom system of the classical predicate logic.

Adding to the mentioned axiom system of the monoidale predicate logic the following axioms (Ml), (M2'), (M3') and (I) gives an axiom system of the general modal logic. Let A and B be arbitrary formulas:

(Ml) U - » A (M2') For each A there is a JB such that A -» B and B -> \B holds. (In the classical case of modal logic we can take A V ->A as B.) (M3') (L4®IJ3) -» \(A®B). (I) —

— (inference rule).

\A —^ \±j

Remark on (Ml): Of course property (PI) of an interior operator is in accordance with axiom (Ml). Remarks on (M2'): In each of the examples of 6.1, ..., 6.4 we have that a •—- — a < int(a •—• —a) only holds for a = 0 and a = 1. Hence axiom (M2) of the classical modal logic cannot be fulfilled in every case of the general modal logic. However in each of the examples of 6.1, ..., 6.4 we have that for every a e [0,1] we have a < 1 and 1 = intl. Remark on (M3') = Each interior operator on a residuiertes ilc-monoid has the property (P3), which is in accordance with axiom (M3'). References [1] U. Hohle, Monoidal logic, in: Fuzzy Systems in Computer Science, Vieweg 1994, 233-243 [2] H. Rasiowa, R. Sikorski, The Mathematics of Metamathematics, Panstwowe Wydawnistwo Naukowe 1963 [3] H. I. Rosenthal, Quantales and their applications, Pitman Research Notes in mathematics 234, Longman, Burnt Mill, Harlow 1990 [4] B. Scapellini, Die Nichtaxiomatisierbarkeit des unendlichen Pradikatenkalkiils von Lukasiewicz, Journal of Symbolic Logic 27 (1962) 159-170 [5] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, 1983

THE STRUCTURE OF AFFINE ALGEBRAIC SETS

ERALDO GIULI Department of Mathematics University of L 'Aquila 67100 L'Aquila, Italy E-mail:giV:li@univaq. it It was shown in [9] that the category SepX of separated affine sets of a hereditary coreflective subcategory X of the category SSet of all affine sets over the twopoint set S = {0,1} admits completions. That is: there exists an epi-embedding reflective subcategory CSepX. of SepX. such that, if / : X —» Y is an epimorphism and an embedding in SepX with Y £ CSepX then Y is the CSepX-reflection (= completion) of X. Here, for each hereditary coreflective subcategory X of SSet, an internal characterization of the complete affine sets (also called affine algebraic sets) of SepX is established and several examples are provided. The obvious statements valid, for example, for the completion of metric spaces as "finite implies complete" and "every set admits a complete structure" are shown to be false in this general contest. Mathematics Subject Classifications 18A30, 54D10, 54D30

(2000): 54A05, 54B30, 54E52,

Keywords: Affine set, Zariski closure, separated affine set, complete affine set = affine algebraic set, one-point extension.

1. Introduction

Let Met be the category of metric spaces and non-expansive maps and let CMet be the full subcategory of complete metric spaces. It is well known (see e.g. [8]) that: (1) for every metric space X there exist a complete metric space X* and a dense embedding 7 : X —» X* such that, for every complete metric space Y and non-expansive map / : X —> Y there exists a (unique) non-expansive map /* : X* —> Y for which 7 o /* = /, and (2) if / : X —> Z is a dense embedding into a complete metric space Z then Z coincides, up to isometries, with X*. Since in Met dense non-expansive maps are epimorphisms, (1) says that CMet is epi-embedding reflective in Met. It is also well known that the category of compact Hausdorff spaces is epi-embedding reflective in the

113

114

GIULI

category of Tychonov spaces, being here X* the Stone-Cech compactification of X. However no property (2) is fulfilled by the latter construction. In this sense we can say that property (2) distinguishes completions from compactifications (see [3] and [2] for a general treatment of categorical completions) . On the other hand no non trivial epireflective subcategory of Top consisting of Hausdorff spaces (e.g. the category of Tychonov spaces) admits completions (cf. [3] Example 1.8(2)) In [9] the (topological) category SSet of affine sets over the two-point set S — {0,1} and a closure operator in SSet, called Zariski closure, were introduced and investigated. It was shown that a class of separated - with respect to the Zariski closure - affine sets admits completions whenever it is the class of all separated affine sets of a subcategory of SSet which is stable under affine subsets, disjoint unions and quotients (i.e., of a hereditary coreflective subcategory of SSet). Examples include the class of all TO topological spaces, in which the complete objects are the sober TO spaces (every irriducible closed set is the closure of a unique point) and the class of all TO closure spaces in which the complete objects are the sober TO closure spaces (every nonempty closed set is the closure of a unique point). In this paper, for every hereditary coreflective subcategory X of SSet, an internal characterization of the complete affine sets of SepX. is established and several examples are provided. Every separated Zariski-compact (in the sense of [5]) affine set is complete but the converse is in general false (e.g. X = Top). Recently Veerle Claes and Eva Lowen-Colebunders gave sufficient conditions on X for the equality: Zariski-compact separated = complete [4]. For categorical terminology see [1] and [13]. For General Topology we refer to [8]. 2. Separated affine sets and completions Recall from [9] that an affine set over the two point set S — {0,1} is a pair (X,U) , where X is a set and U is a subset of the power set P(X). An affine map from (X, U] to (Y, V) is a function / : X —» Y such that f~l(V) € U for every V e V. SSet will denote the category of affine sets (over S) and affine maps. The functional isomorphic description of SSet is as follows: objects are pairs (X, A) where X is a set and A is a subset of the power set Sx and the morphisms from (X, A) to (Y, B) are functions / : X -> Y such that J3 o f £ A whenever (3 G B.

AFFINE ALGEBRAIC SETS

115

SSet with the forgetful functor U : SSet —> Set is a topological category in which the fibres are sets but in which the empty set has two structures and the one-point sets have four structures. An injective and initial affine map is called, as usual, embedding and a subset M of (X,U) endowed with the initial structure induced by the inclusion will be called affine subset of (X,U). Affine sets coincide with so called normal (Boolean) Chu spaces introduced by Vaughan R. Pratt as a generalization of Nielsen, Plotkin and Winskel's notion of event structure for modelling concurrent computation [12]. Moreover continuous maps between normal (Boolean) Chu spaces coincide with the affine maps. Thus SSet is a full subcategory of the category Chus of (Boolean) Chu spaces and continuous maps. Every closure (hence topological) space, is an affine set and a function between closure (resp. topological) spaces is continuous if and only if it is affine. Thus both the categories Top of topological spaces and Cl of closure spaces are fully embedded in SSet. Let M be a subset of an affine set (X,U). The Zariski closure, of M in (X,U) (cf. [9]) is defined by Z(X
= {x e X : (J U,VeU)(Ur\M

=

VnM,xz(U^V)}.

The Zariski closure (or z — closure) is extensive, monotone and continuous but the crucial additional property is the hereditariness (the z-closure of a subset M of an affine subset (Y, V) of an affine set (X, U) is the intersection with y of the z-closure of M in (X, U)). The terms z-closed embedding (resp.: subset), z-dense subset (resp.: affine map) have the usual meaning. Let X a full and isomorphism closed subcategory of SSet. An affine set (X,U) in X is called (1) (z-)separated if the diagonal AX = {(x,x) : x € X} is z-closed in the square X2. (2) absolutely z-closed if it is separated and it is z-closed in every separated affine set of X in which it can be embedded. (3) z-compact if, for every affine set (y, V) of X, the projection p : (X,U) x (y,V) -> (y,V) is z-closed. That is: p sends z-closed subsets into z-closed subsets. It was shown in [9] that SepX admits completions whenever X is a hereditary coreflective subcategory of SSet (Theorem 5.4) and that the class of complete affine sets coincides, in this case, with the class of all absolutely z-closed affine sets of SepX (Proposition 5.1).

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GIULI

A general method to produce hereditary coreflective subcategories of SSet was considered in [9] and goes back to a paper of Diers [7] (from which some terminology is derived, e.g., affine set, Zariski closure, etc.). Recall that an algebra structure in the set 5 = {0,1} is a family of functions n = {wT : ST -> S]

where T runs in a given class of sets. Then for every set X, by point-wise extension, the powerset Sx carries an algebra structure. If Q is an algebra structure in the two-point set S we denote by SSet(J7) the subcategory of SSet consisting of those affine sets (X, A] for which A is a Q-subalgebra of the function algebra Sx. For every algebra structure fi in S the corresponding subcategory SSet(fi) of affine sets over the algebra (S,fl) is hereditarily coreflective in SSet. In this way, for suitable fl, we obtain among others, the topological categories CS of closure spaces, Top of topological spaces and PrOS of pre-ordered sets. A second general method to produce hereditary coreflective subcategories of SSet goes back to [10] where in particular two proper classes of examples were provided: (a) For each infinite regular cardinal a, the class Tight (a) of all topological spaces for which every point in the closure of a subset is also in the closure of a smaller subset of cardinality less than a. If a is a successor cardinal we obtain the TO spaces of tightness less than a; (b) For each infinite cardinal a, the class Alex(a) of all topological spaces for which the intersection of less than a open sets is open. The question whether every hereditary coreflective subcategory of SSet is of the form SSet(f2) has negative answer as point out to the author by Eva Lowen-Colebunders (the class of discrete affine sets form an hereditary coreflective subcategory which is not of the form SSet(f2)). Moreover, while it is clear that, for every a, Alex(a) is of the form SSet(fi) we do not know if there exists an uncountable a such that the category Tight(a) is of the form SSet(fl). 3. Internal characterization Let (X,U] be an affine set and let a be a subset of U. Definition 3.1. The one-point extension of (X,U) determined by a is the affine set (Xa,Ua) where Xa = X U {oo}, oo £ X, and ) ( J { A ^ {00} : A 6 a}.

AFFINE ALGEBRAIC SETS

117

Note that (X,U) admits 1\u\ different one-point extensions. Lemma 3.1. (Subobject) Always (X,U) is an affine subset of (Xa,Ua}. Proof. Directly follows from the definition of Ua.

O

Lemma 3.2. (Density) Always oo 6 z^xa,u^)X Proof. V H X = W H X with V, W 6 Ua implies either V, W 6 U \ a, so that V = W, or V = V' U {oo}, W = W' U {oo}, so that (V' = W', hence) V = W. D A subset a of W is called fixed if a = Ux for some x e AT, where Wx denotes the family of all elements of U containing x. Lemma 3.3. (Separation) If (X,U} is separated then (Xa,Ua) is separated iff a is not fixed. Proof. (Xa,Ua) is not separated iff there exists x G X with {Ua)x = (Ua)x and this is equivalent to Ux = a in (X, U). D Theorem 3.1. A separated affine every subset ofU is fixed.

set (X,U)

is complete in SepSSet iff

Proof. =£-. Assume not, and let a C U be not a fixed subset. Then by Subobject Lemma, the Density Lemma and the Separation Lemma ("if" part) the inclusion (X,U) <—> (Xa,Ua) is a z-dense embedding in a separated affine set, consequently (X,U) is not complete. <£=. Assume not. Since the Zariski closure is (hereditary, hence) weakly hereditary we may assume that there exists a separated affine set (X U {oo}, V) such that the inclusion is a dense embedding. Set

a = {U£U : U i V} Since (X,U) is an affine subset of (X U {oo}, V) we have U U {00} 6 V whenever U € a. On the other hand, by X z-dense in (XU{oo}, V) we derive U € a whenever U U {00} 6 V. In conclusion U e a iff U U {00} € V. This means that (X U {oo}, V) is the one-point extension of (X,U) determined by a. Finally, since (Xu{oo}, V) is separated, by Separation Lemma ("only if" part), we have that a is not a fixed subset. D

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GIULl

Let X be a hereditary coreflective subcategory of SSet and let (X,U) 6 X. Definition 3.2. A subset a of U is called X-compatible if the one-point extension (Xa,Ua) belongs to X. With a proof that follows the lines of the proof of Theorem 3.1 we have Theorem 3.2. Let X be an hereditary coreflective subcategory of SSet. Then (X,U) € SepX. is complete in SepX. iff every ^.-compatible subset of U is fixed. Corollary 3.1. If (X,U) is SepX-complete then the cardinality of X coincides with the cardinality of the set of all It-compatible subsets. Proof. Directly follows from Theorem 3.2.

D

Remark 3.1. The question whether, for every X and for every (finite) set X there exists a SepX-complete structure (at least the discrete one) has negative answer: the empty set has two separated SSet-structures but none of them is complete in SSet. See also the examples below. Example 3.1. (1) Let (X,U) be an affine set. Every subset ofU is SSet-compatible. The affine two-point set ({0,1}, {{!}}) is SepSSet-complete so that every set of cardinality 2'A' admits a complete structure, the one of SA. On the other hand if (X, U} is complete then the cardinality of X is 2 |w l by Corollary 3.8 . No separated affine set containing in the structure, at least, one of the two trivial subsets is complete in 5epSSet. (2) Let SSET be the subcategory of SSet consisting of all affine sets (X,U) containing in the affine structure the trivial subsets. Then a subset of U is SSET-compatible if and only if it contains X and does not contain the empty set (as point out to the author by David Holgate). The affine two-point se£({0,l},{{0,l},0,{l}}) is SepSSET-complete and except for the empty affine set the existence of a complete structure is restricted to the separated (X,U) for which the cardinality of X is of the form 2l A l. (3) Let SSet(C) be the subcategory of SSet consisting of all affine sets (X, U} such that U is stable under complements. Then a subset a ofU is SSet(C)- compatible if and only ifU — a consists precisely of the complements of the elements in a. In particular if (X,U) has n

AFFINE ALGEBRAIC SETS

119

points and it is SepSSet(C)-complete, then n = 2fc with k = U\2. Conversely every finite set of cardinality of the form 2fe admits a complete structure, the one of the k-power of the affine two-point set ({0,1}, {{0}, {!}}) which is SepSSet-complete. An easy computation says that a finite discrete affine set is complete in SepSSet(C) if and only if it has not more than two points. (4) An affine set (X,U] is called closure space if the affine structure contains the whole set and it is stable under arbitrary unions. For a categorical study of the corresponding category, denoted by CS, see e.g. [6]. A nonempty subset a ofU is CS-compatible if and only if the empty set is not in a, it is closed under extensions, that is: if U £ U contains a member of a then it is in a and, U — a is closed under arbitrary unions. Consider for a given nonempty closed set F of(X,U) the family ap of all elements ofU intersecting F. Then it is clear that a subset ofU is CS-compatible if and only if it is of the form ap • This gives directly the known characterization of the complete closure spaces: a separated closure space is complete if and only if every nonempty closed set is the closure of a point (cf.[6]). (5) In the category Top of topological spaces the separated objects are the TQ topological spaces and it is well known that the complete objects are the sober TO spaces (topological spaces in which every irriducible closed set is the closure of a unique point). The above characterization follows directly by the fact that a subset of open sets of a TQ space is Top compatible if and only if it is an open filter such that U - a is stable under arbitrary unions. (6) In the category Tight(a) a subset a of U is compatible if and only if it it satisfies the conditions of the previous example and: there exists a subset of X of cardinality less than a intersecting every member of a.

References [1] J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, Wiley and Sons Inc., (1990). [2] G. C. L. Briimmer and E. Giuli, A categorical concept of completion, Comment. Math. Univ. Carolin. 33 (1992), 131-147. [3] G. C. L. Briimmer, E. Giuli and H. Herrlich, Epireflections which are completions, Cahiers Topologie Geom. Diff. Categ. 33 (1992), 71-93. [4] V. Claes and E. Lowen-Colebunders,Productivity of Zariski-compactness for constructs of affine sets, Manuscript.

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[5] M. M. Clementine, E. Giuli and W. Tholen, Topology in a category: compactness, Portugal. Math. 53 (1996), 397-433. [6] D. Deses, E. Giuli and E. Lowen-Colebunders, On complete objects in the category ofTo closure spaces, Applied General Topology 4 (2003), 25-34. [7] Y. Diers, Affine algebraic sets relative to an algebraic theory , J. Geom. 65 (1999), 54-76. [8] R. Engelking, General Topology, Heldermann Verlag, Berlin (1988). [9] E. Giuli, On classes of TQ spaces admitting completions, Applied General Topology 4 (2003), 143-155. [10] E. Giuli and M. Husek, A counterpart of compactness, Boll. Un. Mat. Ital.(7) 11-B (1997), 605-621. [11] H. Herrlich and R. Lowen, On simultaneously reflective and coreflective subconstructs, Proceedings Symposium on Categorical Topology UCT 1994 (1999), 121-130. [12] V.R. Pratt, Chu spaces and their interpretation as concurrent objects , Springer Lecture Notes in Computer Science 1000 (1995), 392-405. [13] G. Preuss, Theory of Topological Structures, D. Reidel Publishing Company, (1988).

A TWISTED TRIPLE CATEGORY OF TRACK COMMUTATIVE CUBES K.A. HARDIE Department of Mathematics and Applied Mathematics University of Cape Town 7700 Rondebosch South Africa E-mail : hardiekaQiafrica.com K. H. KAMPS Fachbereich Mathematik Fernuniversitdt Postfach 940 D-58084 Hagen Germany E-mail : [email protected] We construct a category whose elements compose horizontally, vertically and laterally. The elements are cubical diagrams of Hausdorff spaces with semitracks in their faces and inhabited by 2-tracks. The compositions are associative but the 2 x 2 properties and the 2 x 2 x 2 property are only satisfied in a twisted sense. Mathematics Subject Classifications (2000): 55Q05, 18B05, 18D05. Keywords: Triple category, semitrack, 2-track, homotopy commutative cube. 0. Introduction

The notion of track homotopy commutative square (a square of spaces and maps that commute up to homotopy, inhabited by the relative homotopy class of a homotopy, i.e. by a track) has been rather influential. As pointed out by Gabriel and Zismann [GZ], with such structures the category of spaces becomes groupoid enriched. On the other hand, horizontal and vertical composition of such squares gives rise to a double category [S W], and as observed by Spencer [S], the result is appropriate for studying properties of homotopy pushout and homotopy pullback. Further connections to Dold's theorem on fibre homotopy equivalences [D], to the track homotopy categories over and under fixed spaces, to the category of homotopy pairs are indicated in the book of Kamps and Porter [KP]. Moreover the concept offers an entree to study of secondary and higher order homotopy composition operations [HJ], [HKK2]. The question has long been asked whether there exists a useful generalisation of these ideas and techniques to three (and higher) dimensions. In 121

122

HARDIE AND KAMPS

[HKK1] a concept of 2-track and a corresponding notion of track commutative cube has been developed in such a way that the category of pointed Hausdorff spaces admits a 2-groupoid enrichment with respect to the closed monoidal structure on the category of 2-groupoids given by the Gray tensor product [Gy]. It was shown that 2-track homotopy commutative cubes can be composed but the authors did not pursue there the question of the algebraic properties of the composition. In part this was due to a perceived difficulty with regard to the nature of the composition restricted to the faces of the cubes, which had to carry semitracks : it was known that semitracks did not satisfy a 2 x 2 property. The purpose of this paper is to demonstrate that a structure related to that of a triple category is indeed obtained. To understand the nature of the 'twisted' triple category we have in mind, we need to recall that a triple category is simply a formal structure appropriate for performing (a certain kind of) 'three-dimensional algebra'. Brown, in his pioneering article [B], explained that elements of a onedimensional algebra may be composed when placed side by side along a line. Elements of a two-dimensional algebra compose also vertically. In a three-dimensional algebra, they compose further in a third dimension, perpendicular to each of the others. Note that the 'elements' themselves (in this good intuition) have the same nature, however they are being composed. Specifically, the underlying double categories of a triple category have the same morphisms as the triple, we simply restrict ourselves to two of the compositions. Following this line of thought, the elements of the twisted triple category are cubes of maps (between Hausdorff spaces) whose faces are inhabited by semitracks, and whose interiors are inhabited by 2-tracks (exactly as in section 4 of [HKK1]). Moreover the horizontal, vertical and lateral composition operations for our structure are all defined there. However, we have to take care with regard to the faces of composite cubes since pasting of semitracks does not satisfy a 2 x 2 property. Indeed to make the operations comparable that would normally be found on either side of the 2 x 2 equations we have to insert a type of twist operator that acts by pasting (to the 4-composite faces) an additional pair of cubes inhabited by appropriate interchange 2-tracks. Although stated in terms of the category of pointed Hausdorff spaces our results will hold in the general setting of a 2-groupoid enriched category. Applications have been postponed to subsequent publications.

TRACK COMMUTATIVE CUBES

123

The authors gratefully acknowledge suggestions by R. Brown, T. Porter, R.W. Kieboom and P.J. Witbooi. 1. A 2-groupoid enrichment of Top* In this section we recall details of the 2-groupoid G2(y, X) associated with the pair of pointed Hausdorff spaces V and X [HKK1]: the 0-morphisms are pointed maps from V to X, the 1-morphisms are semitracks (equivalence class of homotopies under a relation finer than arbitrary relative homotopy), 2-morphisms are 2-tracks (equivalence classes of relative homotopy classes of 2-homotopies). For each 1-morphism (ft) there is an isomorphism a ( f t } : G2

(1.1) which reduces to

if /o — * : V —> X. Moreover, G2(V, X) is a bifunctor, covariant in X and contravariant in V: given pointed maps k' : V —> V and f : X —> X' there are associated 2-groupoid morphisms (k'}- : G 2 (V, X) -» G a (V , X)

/. : G 2(V, X) -> G2(V, X').

and

Furthermore, the 2-dimensional structure comprising pointed spaces, pointed maps and semitracks admits two horizontal composition operations •< and >. Semitracks (gt) •< (ft) and (gt) ^- (ft) are defined in the situation /i

91

v "KM" z li^r x /o

9o

as follows. (1.2)

(gt) 4 (ft) = gi.(ft) • fo(gt) and (gt) > (ft) = /f^t) • 9o.(ft) •

These compositions are associative. They endow the spaces, maps and semitracks with the structure of a sesquicategory [St]. The interchange law not being valid in general for semitracks, the compositions are distinct. However, the compositions are linked by a natural isomorphism \gt,ft]:(9t)4(ft)=*(9t)>(ft) obtained by considering the interchange 2-track.

124

HARDIE AND KAMPS

Suppose that ft : V —> Z and gt : Z —> X are homotopies. Then there is an associated 2-track, an interchange, depending only on {/<) and (gt) and inhabiting the diagram , gtfo , gift 5o/o -*• 5i/o (1-3)

[5t,/tU

Let f t , s be a square in a Hausdorff space W, i.e. a map 7 x 7 —> W . Then a representative of the lens collapse 2-track (1-4)

{ft,.} '• • (/t,o) =» •

of / tia in 62^ is suggested by the following sketch. The construction is due to Grandis [Gs].

(0,0) -- • -- (1,0)

-> w (0,1)

^--

(1,1)

The outer square is first retracted on to the inner square by collapsing the triangles vertically on to its edges. The inner square is then mapped to W by applying ft,3. The interchange 2-track [gt, ft] is then the adjoint of the special case of {ft,s} in which W = Xv and ft,s is the adjoint of the composite 2homotopy gtfs '• V —> X. 1.5. Proposition. [HKK1, Theorem 3.5] The 2-track [gt, ft] eG 2 (V,X) ((9ift • g t f o ) , (flt/i • g o f t ) ) depends only on the classes (gt) and (ft). The following properties hold.

(i) (H) (Hi) (iv) (v)

For a constant homotopy ct, [
TRACK COMMUTATIVE CUBES

125

(vi) Given k : V -» V, ft : V -» Z, gt : Z -* X then [gt, ftk] = k-[gt, ft}. (vii) Given ft : V -> Z, gt : Z -> X, /i : X -> X' toen [%, /t] = h.[5t, /t]. (umj Given /t : V -» Z, g : Z -> X, ftt : X -> VF t/ien [/i t ff, /t] = [/it, 5/t]^ bt./t]"1 = -\9i-t, f i - t ] • (x) If fo = f i = * and go = gi = * are trivial maps then [gt, ft] =0. (xi) Given 2-tracks " = (9t} => <^> : 5 ^ 5' : 5 ^ C

i/ien mG2(A,C)((g'ft.gtf),(g'tf>.gf{))

.

2. Composition of cubical diagrams

A typical morphism of the prospective triple category will be a cubical diagram -*- A2

X

Bl

IX

>- 82

(2.1) Ci

I

x J Di

^C 2

I

xi

>- D2

of spaces, maps, semitracks (occupying the faces) and inhabited by a 2track a\. The map from the vertex space A\ to the space B\, for example, is denoted by A\B\. A composite map, such as C\D\ o AiC\, is denoted more economically by A\C\D\. The semitrack occupying the left hand face is understood to be the semitrack, denoted (C\B\) of a homotopy AidD! ~ A1B1D1 . Note that the positive direction for homotopies between partial routes from AI to £>2 is that of a counter-clockwise rotation.

126

HARDIE AND KAMPS

The cube (together with its inhabiting 2-track) can also be represented by the hexagonal diagram of semitracks \AiC\2D2 Ai2C2D2

(2.2)

In (2.2), opposite and parallel edges correspond to opposite and parallel faces of the cube. Note that expressions for maps such as A\C\GiD-2 are abbreviated A\Ci^D^ and the vertices of the hexagon each denote composite maps from A\ to D^. Horizontal composition of such cubical morphisms has been described in [HKK1], we recall the definition. Suppose we have another whose domain face, i.e. left hand face, coincides with the right hand face of (2.1) : A3

A2 B2

A23C3D3

B3

(C2B2){ C2 D2

D3

At least in the case of the cube diagrams, we have a clear idea that after composition we should end up with a diagram of form

-^- B2

(2.3)

. C2

y

, 03

-*- D 2

with pasting of semitracks occurring along the dotted lines. One might initially think that the hexagon equivalent of (2.3) would be the diagram A12C23D3 A1C1D123

(2.4)

A123C3D3

TRACK COMMUTATIVE CUBES

127

but, on examining the left hand hexagon of (2.4), we observe that the semitracks (DiB2) and (BiA2) that are shown as successive semitrack arrows are not the semitracks that are required to be pasted in the front face of the cube (2.3). These are (DiB2) and (D^Bs), note that they occupy parallel edges of (2.4). When pasted, as in (2.3), an interchange of order occurs, resulting in the creation of an interchange 2-track. In consequence a pair of rhombi has to be added to the diagram as in the following. (2.5) (D2C3)AlC™D3

(CtAi)

A pleasant feature of the process of pasting of hexagons is that hexagon edges that are parallel correspond to parallel faces of the pasted cubes. As discussed in [HKKl; (4.6)], the interchange process generates the 2tracks (2.5.1)

N2 = {(D2C3), (dA2)}

S2 = -[(D2B3), (BtA2)} .

Then the outside of the composite hexagon (2.5) can be seen to correspond to the outside of the composite cube (2.3), and the composite 2-track inside (2.5) is defined to be the 2-track inhabiting the interior of (2.3). Similarly, two cubes can be composed laterally if the front face of the first coincides with the rear face of the second. The reader will observe that lateral composition of cubes corresponds to pasting of hexagons along common slant edges of negative slope (and adding appropriate rhombi to complete a hexagon). Likewise, vertical composition of cubes is possible if the bottom face of the first cube coincides with the top face of the second, and then their associated hexagons are pasted along common slant edges of positive slope. We shall see later that interchange tracks will also need to be invoked when pasting of cubes in more than one direction is contemplated. For example,

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HARDIE AND KAMPS

there is not a unique way of pasting semitracks in a composite face such as A -*- B -»- C

K.DB>1t\<EC>K\

I

!

I

D...........E...........F

(2.6)

I G -*- H ->• K

as can be seen from the following routings. (GE) (HF) ADGHK ->• ADEHK ->• ADEFK

(DB)

(DB)

ABEHK ->- ABEFK -*- ABCFK (HF) (EC)

2.7. Remark. More formally, the hexagonal diagram can be viewed as a square in the double category associated to the 2-groupoid Gz(A-\.,A^}. The composite (2.5) can be interpreted as a composite of squares in the double category associated to the 2-groupoid Ga (-4 1,1)3). Alternatively, we refer the reader to the general treatment of pasting in a 2-category given by Street in [St, section 3].

3. A twisted triple category

Brown and Higgins [BHl] define a multiple category to be a set with a family of category structures any two of which satisfy an 'interchange' law. In the case n = 2, the additional property is sometimes referred to as the middle four interchange. Essentially correspondingly, we regard a triple category as a class of morphisms with three independent composition operations, any two of which enjoys the structure of a double category, and in which the structure as a whole satisfies a middle eight interchange property. We think of the structure as being 3-dimensional and, in particular following Brown [B], with the three compositions being performed in three mutually perpendicular directions, i.e. horizontally, vertically and laterally. Accordingly we use the symbols 0, 0 and ® to denote these operations.

TRACK COMMUTATIVE CUBES

129

To assist interpretation of these, and since we also wish to suggest pasting of hexagons, we display these in the form

a © 0 ,

7 0 , S

A ® f.

when the relevent compositions are defined. Each composition is required to be associative, the class of morphisms must be replete with identities for each composition and the following 2 x 2 (i.e. middle four interchange) properties must hold. 72 © 7i

(3.1)

e

oti © ai

(3.2)

72

7i

"2

"i

= e © e

=

«2 © Oil

OC2

Si

e

(3.3)

3, 71

e

fa

<8> =

Ti

Si

0

<8

ft

It is also necessary that a 2 x 2 x 2 property (middle-eight interchange) holds, however, it is well known that this is in fact implied by the above conditions (cf. [BH2, p. 238]). We wish to consider the equations (3.1), (3. 2) and (3.3) in the context of the following 2 x 2 x 2 diagram of cubes, regarding each of the 8 subcubes as being morphisms [a$], [fa], [74], [<5j] (i — 1, 2) to be composed, where we understand that [a4] is the cube with domain vertices Ai, [fa] the cube

BARDIE AND KAMPS

130

with domain vertices Bi et cetera. *- A2 -

Ai

-*• A3

-^B 2

\ ^J

2

1

(3.4)

"

\ \

-*. \

-*• r2 \

F 1

XI _

-*G2 Ji

-^Ja

Recollect that ©, 0 and <8> refer to the operations of horizontal pasting, vertical pasting and lateral (slant) pasting respectively. In consequence, note that [a^] © [«i] corresponds to a cubical diagram of the form (2.3) situated on the upper level at the rear of (3.4). On the other hand, note too that the left and right hand terms of equation (3.1) both correspond to the rear vertical slice of (3.4). However, the difference in the order of pasting will imply that the semitracks occupying the composite faces will not be equal in general in view of the effect pointed out in (2.6). To make the two pastings comparable we shall have to insert a twist operator (indicated by a T) to one or other of the sides of (3.1). Such a twist is performed by pasting cubes inhabited by interchange 2-tracks to each of the four-fold composite faces (i.e. we conjugate with the relevant interchange cube). Note, however, that these composite faces occupy a plane perpendicular to the C^DI direction. Alternatively put, the axis of the twist T is the C-^Di direction. When cubes are pasted there are also twist operators a and p with axis directions B-^Di and D\D^ respectively. Thus we come to the following 3.5. Definition. A twisted triple category (T, ©, Q, together with natural isomorphisms T, a and p (twist operators) satisfying 72

(3.5.1)

e

TRACK COMMUTATIVE CUBES f a ® Pi

(3.5.2)

P i "

fa

OL
131

a2

©

01 g,

(3.5.3)

®

=

71 © o-i

(71 0 1r 01 J

I

ai

®

as well as a twisted form of the middle eight interchange property: A = Ba» = CP°T ,

(3.5.4)

where A, B and C (called snowflakes) are as follows. 62 © 5i ©

(3.5.5)

A

=

®

02 ©01

72 ® 7i © C*2 © a l

<52 <8>

61 ©

72 (3.5.6)

S

=

(3.5.7)

C

=

®

7i ©

72

7i

©

©

©

132

HARDIE AND KAMPS

3.6. Theorem. With cube morphisms and compositions as described above there are twist operators T, a and p with respect to which we have the structure of a twisted triple category. As remarked above, the 2 x 2 x 2 property in the context of a strict triple category is a consequence of the three 2 x 2 properties. We shall see that a similar result holds for twisted triple categories: 3.7. Proposition. With the hypotheses of Theorem 3.6, the equations (3.5.4) are o consequence of the twisted 2 x 2 properties (3.5.1), (3.5.2) and (3.5.3). Proof By (3.5.1) we have

S2 © 61

e

02 © 0!

A

=

W"

=

<8> {M}r

72 © 7i

e

OL-2 © ai 72 7i <^2 <5i where p, = Q © © and A = © © © . However, if we recall that the a2 ai /32 f3l twist operator T functions by pasting cubes containing interchange tracks, it is clear that when the twisted versions of A and \JL are pasted (via (g>) the interior interchange cubes cancel so that we find

{A}T A =

®

(

= \

A

r\

.

But now we may expand the right hand side by (3.5.2) to obtain

AA

- \

I 72 © CC2

®©

71 © Oil

TRACK COMMUTATIVE CUBES

133

Finally, applying (3.5.3) we find

72

A =

7i

e

however, applying an argument given above but in reverse, we may reinsert cancelling p twists in the middle to achieve

7i

72

A =

©

cpaT .

©

On the other hand via (3.5.3) and then (3.5.2), we have 52 © 81 f 72 © 7i

A = I

62 72

e

71 ©

- Bap .

/?2 © 0! Q!i

3.8. Remcirk. The reader will note that the composition operators A, B and C are characterised by the order of their constituent operations ©, 0 and ®. We may express them as formal products :

and the twist operators as involutions. The relations between these and the other three composition operators (snowflakes) are indicated by the diagram

e©(

•e®©

134

HARDIE AND KAMPS

4. The 2 x 2 properties We begin with a verification of (3.5.1), noting that 0:2 © ai has been represented as a hexagon in (2.5). To evaluate the left hand side of (3.5.1) we observe that

7l

=

C1F1G12

C12F2G2

lDl>| C:D1G12

^ |(F2

C 2 F 2 G 23 72

=

C12D2G2

C 23 F 3 G 3

°2>| C2D2G23

G23D3G3

X^

'2D23G;

;«2G3>

and obtain the hexagon representing 72 © 71 :

C12F2G23

Cl 2 3 F 3 G 3

(4.1) G] 2 3 D 3 G 3

Next we observe that the bottom right slant edge of the hexagon 4.1 coincides with the top left slant edge of the hexagon 2.5 so that the two hexagons can be pasted. In this way we obtain a (larger) hexagon corresponding to of form : (4.2)

A1C12D2G23

A12B2D23G3

TRACK COMMUTATIVE

135

CUBES

The interchange tracks present have been labelled. A similar diagram is obtainable for the right-hand side of (3.5.1). We first obtain (GlF2>

A12C2F2G2

(G1D2)

A 2 B 2 £) 2 G 23

and then (4.3)

A 12 C 2 F 23 G 3

yl123C3F3G3 Ai 2 B 2 D 2 G 23

136

HARDIE AND KAMPS

Note that the semitracks on the boundaries of (4.2) and (4.3) do not appear in exactly the same order. To be specific, in (4.2), on the top right edge the middle pair of semitracks (F^Cs) and (C7iA2) do not appear in the same order as they appear on the top right edge of (4.3). A similar remark can be made regarding the middle pair of semitracks on the lower left edge. However this was expected and the twist has yet to be performed. In order to achieve the desired twisted 2 equality we insert an interchange rhombus above the top right edge of (4.2) to correct the first defect and another interchange rhombus below the lower edge of (4.3) to correct the second defect. We shall see that this is exactly what is required. Examining diagrams 4.2 and 4.3 for common features, we notice that the 2-tracks along the top left edges, N I , 71, Y\, respectively along the lower right edges, 62, 0:2, X% coincide. Moreover the remaining central portions have two sections. The corresponding upper sections being

AiCi 2 F 2 G 2 3 I

A1Ci23F3G3 I\ • Ai 2 G 2 3FsG 3

AiCi 2 F 2 G 23 I\

A12(72F23Ga /* \\F^C3)

AiGi 2 .D2G 2 3 •

Ai 2 G 23 F 3 G 3

Ai2C2D2G23

Inserting interchange 2-tracks into the second subdiagram we obtain the following.

(CiA 2 )

(G 2 £> 3 )

TRACK COMMUTATIVE CUBES

137

With this modification to the second subdiagram the two are indeed comparable since their boundary semitracks coincide. Moreover, we claim, the 2-tracks in the interiors are equal for the following reason. Firstly their hexagonal parts are both induced by 72 and, by examination, we may observe that the difference of their interchange components is given by the following pasting of interchange rhombi. (F3D3)

(F3D3)

However we have the following. 4.4. Coherence Principle. Interchanging a fixed semitrack with the bounding semitrack of a (homotopy commutative) cube hexagon yields a zero 2-track. The following proof, although given for our special case is quite general. We are in the situation

where £ denotes the result of pasting the semitracks around the hexagon containing 72 and (c) denotes the semitrack of the constant homotopy at one of the vertex maps. Then when we apply Proposition 1.5 (xi) with fi = 72 and v an identity 2-track, the contributions containing /j, and v cancel, [(C\A-^, (c)] = 0, by Proposition 1.5 (i), so that [(CiA^), S] = 0. Continuing the main argument, we note that a similar argument can be applied to the two lower sections of the central portions of diagrams 4.2 and 4.3, and hence the equality of their total tracks is achieved modulo inclusion of interchange 2-tracks as described. Moreover the internal insertions cancel so that the only interchange 2-tracks that need to be included to effect equality are those corresponding to the boundary interchanges. Note further that the diagram (4.2) corresponds to the formal pasting of hexagons corresponding to the left hand side of (3.1). Accordingly we propose to use the label (1L) for it and the label (1R) for the right hand side of (3.1). Similarly we use labels (2L), (2R), (3L) and (3R) for the left and right hand sides of (3.2) and (3.3).

138

BARDIE AND KAMPS

Appropriately abbreviated these are as follows.

4/?2

(2L)

(2R)

(3R)

(3L)

In considering the validity of the equation (3.5.2), we may observe that in the above diagrams for (2L) and (2R) the extreme portions (those containing /?i and a-2) are identical and the central portions are related by a shift such as we have seen is permissible in view of the coherence principle. Thus they are equivalent under the permitted semitrack interchange. The situation for (3L) and (3R) is similar. To complete the proof of Theorem 3.6 we need only verify the associativity of the composition operations ©, 0 and ®. To do so, let us envisage composing (2.3) via 0 with another cube

B3

(4.5) C3 D3

TRACK COMMUTATIVE CUBES

139

inhabited by a 2-track 0:3. The corresponding diagram of composed hexagons has a top piece of the form

(4.6)

There are two expressions for the pasted interchange 2-tracks inhabiting (4.6). These are equal in view of the equalities Proposition 1.5(ii),(iii),(viii). A similar remark applies to the bottom piece of the diagram of composed hexagons. The arguments in the case of the compositions 0 and ® are similar. D 5. The six snowflakes We now consider the first snowflake A of (3.5.5). First note that the hexagon corresponding to

e

, can be obtained from the hexagon corresponding

02® 01

to

e

by replacing the letters A, C, F, B, D, GbyB, D, G, E, H, J

c«2®o>i

respectively in the diagrams (4.2) and (4.3). To begin pasting, note that the order of the semitracks on the upper right edge of (4.2) is (5.1)

(FiC2)
(C2A3) .

When letters are systematically replaced in (5.1) as specified above, observe that we obtain exactly the semitracks in the lower left edge of (4.2). In other words we may paste the hexagon obtained onto the bottom left edge of (4.2). If this is done we arrive at the following hexagon which we have abbreviated by replacing objects by bullets, omitting labels of interchange tracks and simplifying the labels of the other 2-tracks.

140

HARDIE AND KAMPS

(5.2) (G2F.

,Y
A

=

' (fflJS2

Labels of semitracks have only been entered on the boundary. However, if a label of some other semitrack is required, one can recover it from the label associated with the hexagon edge of which it is a parallel translate. For comparison we display the hexagons corresponding to © ® 0 (5.3) • _ (FiC 2 >

472

(Giffi)Y (C3B3)

(EiB2)

TRACK COMMUTATIVE

CUBES

141

and to 0 ( (5.4) •(FjC2)

(B2A3)

References [B] R. Brown, Out of line, Royal Institution Proceedings 64 (1992), 207243. [BH1] R. Brown and P.J. Higgins, The equivalence of oo-groupoids and crossed complexes, Cah. Top. Geom. Diff. 22 (1981) 371-386. [BH2] R. Brown, P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981), 233-260. [D] A. Bold, Partitions of unity in the theory of fibrations, Ann. Math. 78 (1963), 223-255. [Gs] M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Cat. Structures, 5 (1997), 363-413. [Gy] J. W. Gray, Formal Category Theory, Adjointness for 2-catego- ries, Lecture Notes in Math. 391, Springer-Verlag, New York, 1976, 1974. [GZ] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, SpringerVerlag, 1967. [HJ] K.A. Hardie and A.V. Jansen, Toda brackets and the category of homotopy pairs, Quaestiones Math. 6 (1983), 107-128. [HKK1] K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space, Appl. Cat. Structures 8 (2000), 209-234.

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[HKK2] K.A. Hardie, K.H. Kamps and R.W. Kieboom, Higher homotopy groupoids and Toda brackets, Homology, Homotopy and Applications 1 (1999), 117-134. [KP] K.H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific, Singapore, 1997. [S] C.B. Spencer, An abstract setting for homotopy pushouts and pullbacks, Cah. Top. Geom. Diff. 18 (1977), 409-430. [St] R. Street, Categorical Structures, in M. Hazewinkel (ed), Handbook of Algebra, Vol. 1, Elsevier, Amsterdam, 1996, pp. 529-577. [SW] C. B. Spencer and Y.L. Wong, Pullback and pushout squares in a double category with connection, Cah. Top. Geom. Diff. 24 (1983), 161-192.

PRODUCTIVITY OF COREFLECTIVE CLASSES IN SOME TOPOLOGICAL STRUCTURES

MIROSLAV HUSEK* Departement of Mathematics, Charles University, Sokolovska 83, 18675 Prague, Czech Republic E-mail: mhusek@karlin. mff. cuni. cz

This paper is partly a survey on behavior of products in coreflective classes of some topological categories and partly it brings some new results completening those known. Mathematics Subject Classifications (2000): 54B10, 18B30, 03E10 Keywords: cartesian product, coreflective class, locally convex space, topological group, topological space, uniform space

We shall start with a motivation for our title and, after defining and explaining some basic notions, we concentrate on productivity of coreflective classes in several topological categories, mainly Top of topological spaces, Unif of uniform spaces, TopLin of topological linear spaces, TopGr of topological groups and their subcategories - we shall refer to them as "our" categories or structures. Before going into a more detailed motivation, I want to stress at this place that my basic motivation for investigation reflective and coreflective subcategories of topological structures came from Horst Herrlich's famous publication Topologische Reflexionen und Coreflexionen, [6].

"The author acknowledges a partial flnancial support of the grants GACR 201/03/0933 and of MSM 113200007. He also thanks for hospitality of Eurandom at TU Eindhoven during writing this paper 143

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1. Introduction Investigating closedness of classes of some topological structures under basic constructions (products, coproducts, subobjects and quotients) is sometimes easy, sometimes difficult. For instance, look at Top. The only subclass closed under all 4 basic constructions is Top itself, [14]; in fact, the only proper classes of topological spaces closed under products, subspaces and quotients, are the class of at most one-point spaces and the class of indiscrete spaces (indeed, all the other such classes must contain a nonindiscrete two-point space and, thus, all zero-dimensional spaces, thus all spaces). On the other hand, except the class of at most one-point spaces and the class of indiscrete spaces, all the other productive and hereditary classes in Top are closed under arbitrary disjoint sums (i.e., coproducts); indeed, S/Xj is a retract of / x II/Xi, where / bears discrete topology, and every our productive and hereditary class contains all discrete spaces (in fact, all zero-dimensional spaces). Unlike trivial situation for epireflective classes in Top explained in the previous paragraph, there are many nontrivial coreflective classes in Top (i.e., classes closed under disjoint sums and quotients) that are or are not hereditary. The class of sequential spaces or of locally connected spaces is not hereditary, the class of P-spaces or of discrete spaces is hereditary. We are not aware of some general rule helping to recognize hereditariness of coreflective classes. From the mentioned 4 corenective classes in the previous paragraph, the last three ones are finitely productive, the class of sequential spaces is not. Since I could not find a nontrivial productive corenective class in Top, I asked at Conference on Categorical Topology in Berlin, 1978, whether such a class exists, [10]. This question was found not to be easy, and is still open in ZFC. Also in other categories it is not easy to deal with productivity in coreflective classes. When we look from a similar point of view at Unif, we find many differences. The first result is the same: The only productive and hereditary class of uniform spaces closed under disjoint sums and quotients is Unif ([8] where that result is shown to be true also for some, but not all, nice subcategories of Top and Unif). On the other hand, there are many nontrivial epireflective classes in Unif closed under quotients, like the class of precompact spaces (or spaces having covering character less than a given cardinal). Consequently, the mentioned epireflective classes are not closed under disjoint sums. There are epireflective classes closed under disjoint

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sums (and, thus, not closed under quotients), like the class of separated spaces, or of zero-dimensional spaces. As for a relation of coreflective classes to subspaces, the situation is similar to that in Top. Clearly, if an epireflective class C in Unif contains a disjoint sum E/Xj of nonvoid spaces Xi, then it contains the uniformly discrete space with the underlying set /. The converse is true in Top but not in Unif. Take an infinite strongly rigid compact space S and the uniformly discrete space N. The epireflective hull of S x N contains N but not a countable disjoint sum Y of the spaces S. Indeed, if we take a countable collection {/„} of uniformly continuous maps on S into 5 x N, then every fn is either constant or an identity map into S x {n} for some n. Consequently, the uniformity on the set Y generated by such maps coincides with that of S x N and differs from the uniformity on Y. In fact, for every infinite regular cardinal K we may take a strongly rigid compact space S having its weight at least K and define C to be the epireflective hull of all uniformly discrete spaces and of all disjoint sums of less than K of spaces S. Then C is closed under disjoint sums of less than K its spaces and not under disjoint sums of K its spaces. We shall see that productivity of coreflective classes in Unif behaves also in a different way than in Top. We shall also mention behavior of products in coreflective classes in topological groups and in topological linear spaces and in some of their subcategories. Surprisingly, the behavior is closer to Top than to Unif although all the morphism there are uniformly continuous with respect to canonical uniformities. First we need to define and explain some basic notions that we shall use often or are not defined in basic books on general topology, topological linear or locally convex spaces, or topological groups.

2. Basic notions Every subcategory will be full and so it suffices to speak about subclasses of objects instead of subcategories. We recall that a subclass C of a category K. is K-productive if every product (in K.) of less than K members of C belongs to C; finite (or countable) productivity is another expression for u>- (or u>i- , resp.) productivity. Productivity number of a subclass C of K. is the smallest cardinal K (if it exists) such that a product in /C of K-many objects from C does not belong to C, otherwise it is a symbol oo that we consider to be bigger than any

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cardinal in this case. Productivity number of C will be denoted by pc or just p (thus, C is pc-productive and not pj-productive). Very often (and it is the case of coreflective classes in our structures) one may take powers of a single space instead of products of spaces in the definition of productivity numbers (take the disjoint or direct sum of the coordinate spaces and realize that the original product is a retract of the power of the sum). Let us recall that sequential cardinal is a cardinal K such that there exists a sequentially continuous noncontinuous real-valued map on the Cantor space 2 K . Those cardinals were dealt with in the classical Mazur's paper [15] and in [17]. Mazur showed that the first sequential cardinal is weakly inaccessible and that every sequentially continuous map on a product of less than K-many metrizable separable spaces into a metrizable space is continuous (even a little more general spaces can be used). Noble [17] generalized the class of metrizable separable spaces used in the last mentioned result to a bigger class including first countable spaces. The first sequential cardinal has its continuation in a hierarchy of similarly defined higher cardinals. We shall define them by means of submeasures: A submeasure n on an algebra B of sets is a real- valued mapping defined on B and having the following properties: = 0, n(B) for AcB,A,BtB, U B) < n(A) + n(B) for A, B & B. A mapping / between topological spaces is said to be T- continuous, for a cardinal T > u>, if it preserves limits of nets of lengths less than T. In algebras of subsets of a set A we shall use the topology inherited from the Cantor space 2A, i.e., a net {Ai} converges to A if limsup{Aj} = liminf{Aj} = A. Definition 2.1. An infinite cardinal K is said to be submeasurable if there exists a nonzero K-continuous submeasure on the algebra of all subsets of K vanishing at singletons. The first submeasurable cardinal equals to a; and the second one coincides with the first sequential cardinal (if it exists - see [11]). Various properties and characterizations of submeasures and submeasurable cardinals can be found in [1] (e.g., characterizations by means of monotone continuity or homomorphisms of groups).

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3. Topological spaces We mentioned in Introduction that it is not yet known in ZFC whether there exists a nontrivial productive coreflective class in Top. The formulation suggests that a consistency is known. It is really the case: Theorem 3.1 (Dow and Watson in [4]) [GCH+no inaccessible cardinal] or [D(A) for all regular cardinals] Every countably productive coreflective class in Top coincides with Top. In fact, the authors assumed the condition "finitely productive and contains a convergent sequence" instead of "countably productive". It follows from the next Proposition 3.1 that both conditions are equivalent for coreflective classes in Top. Theorem 3.2 (Husek in [12]) The class of productivity numbers of coreflective subcategories of completely regular spaces Top coincides with the class of all submeasurable cardinals and {2,00}. The last result implies that every countably productive coreflective class in Top is s-productive and, thus, productive if the first sequential cardinal s does not exist. To prove Theorem 3.2 one needs a characterization of productivity of coreflective classes in Top by means of powers of a testing space: Proposition 3.1 (Husek in [12]) A coreflective class in Top (or any epireflective subclass of Top containing a non -indiscrete space) is Kproductive, K > u, iff it is finitely productive and contains the Cantor spaces 2A for A < K. Are the same results true if we take coreflective subcategories of some productive classes in Top, e.g., of epireflective classes in Top? I conjecture that the answer is in the affirmative. Nevertheless, I can prove that for some cases only. An important case is that of completely regular spaces Topcfl. Since the proof of Proposition 3.1 (and the procedure how to get Theorem 3.2 from it) can easily be modified for coreflective subclasses of any epireflective subclass of Top containing a non-indiscrete space, we get the following half of what we would like to assert: Theorem 3.3. Productivity number of a finitely productive and nonproductive coreflective subcategory of an epireflective subclass of Top containing a non-indiscrete space Top is submeasurable.

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To get the other half, one needs to construct coreflective classes having a given submeasurable cardinal for its productivity number. We can show here how to do that in completely regular spaces (the procedure is a modification of a procedure from [4]): Proposition 3.2. For every submeasurable cardinal K there exists a coreflective class in Topcn with productivity number equal to K. Proof. For a submeasurable cardinal K and a corresponding submeasure H on K, the requested coreflective class C can be described as follows: For a family ? = {fa : a e K} C Ci(X) we define the following real- valued function on X: Tf(x) = inf{r : n{a : fa(x) < r} > 1 - r} . Now, define C C ^opcR as C = {X : Tf(x) is continuous for every /t-indexed family T C Ci(X)} .

4. Uniform spaces Productivity of coreflective classes of uniform spaces was investigated in [9] and [13]. Compare the next result with Proposition 3.1. Proposition 4.1 (Husek in [9]) A coreflective class of uniform spaces is productive iff it is finitely productive and contains all powers of all uniformly discrete spaces. The next result depends strongly on the result from [13] that any product of quotient maps in Unif is quotient. Without that result one would have to take all spaces X from C (as is the case in Top) and not from a generating class. It implies existence of productive kernels (biggest /t-productive coreflective class contained in a given coreflective class) - productive hulls exist in all our categories. Proposition 4.2 (Husek and Rice in [13]) A coreflective class C C Unif is K-productive iff X x D^ £ C for every uniformly discrete space D, A < K and every space X from some subclass of C having C as its coreflective hull. Now we come to the biggest difference between Unif and Top (or

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Theorem 4.1 (Husek in [9], Husek and Rice in [13]) The class of productivity numbers of coreflective classes in Unif coincides with the class of all infinite regular cardinals and {2,00}. Theorem 4.1 is formulated differently in both cited papers than here. In fact, the result is not proved completely there but what remains is very easy. In [9] the corresponding class C having pc equal to a given uncountable regular cardinal K (the case K = LJ is trivial) is the coreflective hull of a strongly rigid compact Hausdorff space S and of its A-powers for A < K (see, e.g., [19] for existence of spaces S). Theorem 4 from [9] asserts that C is not «+-productive; it is easy to show that it is reproductive (one uses a result of Herrlich from [5] that continuous maps SK —> S are either constant or projections). In [13], the corresponding class C is the coreflective hull of special adjacent K-long sequences: C = {X : every uniformly K—sequentially continuous map on X is continuous}. Theorem 4, item 5, asserts that C is K-productive and it is easy to add that it is not «+-productive. We shall complete the summary of the results for Unif by the following one that again does not hold (at least consistently) in Top. Proposition 4.3 (Husek and Rice in [13]) There is a proper productive coreflective class C in Unif. // the sequential cardinals do not exist, thenC coincides with the coreflective hull of metrizable uniform spaces (i.e., of the space W). 5. Topological linear spaces In the category TopLin and in its subcategory LCS of locally convex spaces we shall always assume that coreflective classes C contain R or, equivalently, that C are bicoreflective, i.e., the coreflective maps are linear isomorphisms. In other words, C are closed under inductive generation, i.e., under quotients, direct sums and contain the finest topological linear spaces (or the finest locally convex spaces, resp.). We thus exclude the coreflective class consisting of the zero space and the coreflective class of exotic spaces (having no non-zero continuous functional); both those classes are productive, the former one is also hereditary, the latter one is not hereditary. Since finite products coincide with finite direct sums, every coreflective class is finitely productive. Therefore, all productivity numbers are infinite. In TopLin, productivity numbers pc have one more property, namely no product of at least pc many spaces from C having nontrivial separated

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modifications belongs to C\ that follows from an important result of P. and S.Dierolf [3] that a coreflective class C is K-productive in TopLin iffRx &C for all A < K. Thus we have a testing space for productivity as in the case of Top. In my several talks I claimed that the class of productivity numbers of non-productive classes in TopLin coincides with the class of submeasurable cardinals. Although that result was said to have been published in [11], only a special case was proved there, namely the case for the first sequential cardinal. The original idea of the author was that the published special proof can be almost automatically transferred to higher cardinals. But it is not the case and we shall provide a full general proof here. Theorem 5.1. Productivity numbers of coreflective TopLin. are submeasurable cardinals or oo.

subcategories in

Proof. Let C be a coreflective class in TopLin and K be its productivity number. We may suppose that u < K < oo. We know that there is a noncontinuous linear map / : RK —> E, where E is a Frechet space, such that / is continuous on the coreflection cRK of RK in C. It was proved by Noble in [17] that a map / on a product of spaces into a regular space is continuous iff its restrictions to any canonical image of 1K and to a S-product (in fact, a-product suffices) are continuous. Thus, in our case, either a restriction to a canonical image of 2K is not continuous, or the restriction to the aproduct Y of /t-copies of R at the point 0 is not continuous. We shall show that the latter case is not possible. Otherwise there is an e > 0 such that every neighborhood of 0 in Y contains a point x with |/(a;)| > e (by .[ we denote the distance to 0 in E). Start with a finite set Co and find a point XQ G R* having a finite support C\ such that \ f ( x o ) \ > e and |prj(a;o)| < 1 for i 6 Co- Next, we can find a point xi having a finite support Ca such that \f(xi)\ > e and (pr^Xi)! < 1/2 for i e C\. Continuing in the same way, we get a sequence {xn} of points of Y having supports Cn and such that \f(xn)\ > s and |prj(x n )| < l/(n + 1) for each i e Cn-i and each n. Denote by S the union of all Cn's. Thus, S is countable and Rs canonically embeds into M K . The image contains all points xn. Since the restriction of / to that canonical image must be continuous (because Rw belongs to C) and xn —> 0 we have f(xn) —» 0, which contradicts our assumption. It follows that / is not continuous on some canonical image of 2K into R K . We remind that the canonical image means in our case that there is a point {ra} e RK and a map V = nrei/>a : 1K -> RK with i/>a(0) = 0,^ a (l) = ra. It remains to show that the composition fifr is K-continuous. To simplify

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notation we shall forget about i/j and shall assume that 2K is a part of RK - equivalently, that all ra ^= 0. Take some A < K and a net {xa}aex in 2K converging to 0. Suppose that |/(^a)| > £ for all a G A and some e > 0. For every coordinate /3 e K, the net {pr/3(xQ)}a is eventually 0 starting with an index ap. Therefore, the set K decomposes into A many sets Ba = {(3 e K : a@ = a} (some of the sets Ba may be empty). For y e 2K we may take a special net {ya}ae\ in 2K converging to y, namely prs (ya) = prs (y) if 7 < a and pr B (ya) — 0 otherwise. If { f ( y a } } does not converge to f(y) for some y, we may assume in our case that xa = y — ya. In this case we may define a linear continuous map h = TUia : R A -> n a6A K Ba with /i a (l) = pr Ba (y). Then the net {xx\a}aeA converges to 0 in R A and its /i-image is the net {xa}, which contradicts our assumption that f ( x a ) does not converge to 0. So, we may assume that f ( y a ) converges to f(y) for every y. It follows that we may substitute the original points xa by x^ for some 7 (depending on a) and take e/2 instead of e. It follows we may assume that xa is 0 on IJ{-^<5 : <5 > 7} for some 7. On the other hand, for every 7 6 A the restriction of the net {xa} to (j{Bs : 8 < 7} is eventually 0. Consequently, we may choose a cofinal set S in A such that the supports of xa's are disjoint for different a € S. Now it is easy to follow the construction of h from the previous paragraph and to get a contradiction. Hence, the mapping / on our 2K is K-continuous and not continuous, which means that AC is a submeasurable cardinal. D The next result shows that every submeasurable cardinal is attained as a productivity number of a corefiective class in TopLin. Our procedure is a modification of the proof of Theorem 3 from [11]. Theorem 5.2. For every submeasurable cardinal K there is a coreflective class C in TopLin having K for its productivity number. Proof. Take a K-continuous subrneasure /j, on K vanishing at singletons and having //(«) = 1. Denote by E the linear space R K endowed with the metric X d(x, y)= I , yl . d/j,. J l + \x-y\

Then (E, d) is a topological linear space and the identity mapping / : R K —» E is not continuous since the net {XK}K&[K]<" of characteristic functions of finite sets converges to XK in RK but d(\K, \K) = 1/2 for every finite set

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K. We shall now prove that fg is continuous for every continuous linear map g : RA ->M / t ,A < K. Take a net {x^}/ converging to 0 in R\ Because of weight of MA we may assume that |J| < K. Choose s > 0 and define Ai = {/3 € K : \pr/3(g(xj))\

< e/2 for j > i} .

Then {A,}/ is a cover of K and Ai c Aj for i < j. Since fi is ^-continuous, there is some i0 £ / such that ^(K \ AiQ) < e/2. Then

for j > i0. Consequently, (/(xj) converges to 0 in (E,d). We have proved that the coreflective hull C of {MA}^
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and, conversely, every pseudonorm is a composition of a linear map into a metrizable topological linear space F (or a Frechet space) and of the canonical pseudonorm on F. Proposition 5.1. The following conditions are equivalent for an infinite cardinal K: 1. K is submeasurable; 2. there, is a noncontinuous linear map on R K into a topological linear space that is K-continuous; 3. there is a noncontinuous pseudonorm on RK that is K-continuous; 4- there is a noncontinuous real-valued function on RK that is Kcontinuous. Proof. Clearly, 2 => 3 =>• 4. The implication 1 =>• 2 is shown in the proof of Theorem 5.2. It remains to prove 4 => 1. Assuming 4, we shall show that there is a noncontinuous, K-continuous real-valued function on Z K . Let / : R* —> R be a K-continuous non-continuous function. We want to find a continuous map h : ZK —> R K such that the composition fh : ZK —> R is /^-continuous and non-continuous. By a classical result R is a continuous image of Zw (see, e.g., [16]) by a mapping g, and the composition fgK is thus K-continuous. It remains to show that we can find g in such a way that fgK is not continuous. Since / is not continuous, there is a net {:TJ}/ in R* converging to some point x such that /(a:,) does not converge to f ( x ) . Clearly, we may assume that x — 0 So, it suffices to find g such that for every net {a,}/ in R converging to 0 there is a net {bi}/ in Zw converging to 0 and g(bi) = a^ for every i € /. Take the usual continuous map mi of {—1,0,1} W onto [—1,1] assigning to {en} the point J^ cn/2n}. The subspace P of Zw of all sequences having 2 for their first coordinate is homeomorphic to the space of irrationals and, thus, there is a continuous map m2 of P onto R. The space {—1,0,1} W U P is a closed subspace of Zw and the mapping mi U m? can be extended continuously to a map g : Z^ —> R. Now, every net {«»}/ converging to 0 in R must be eventually in [—1,1]. For those points a^ belonging to [—1,1] we find bi with ra\(bii = a^ and for the remaining points a, we take arbitrary points bi with g(bi) = a^. Then bt converges to 0 and we are done. D 6. Topological groups

We are not aware of results concerning classes of topological groups being closed under all 4 basic constructions. An investigation must start at classes

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of abstract groups (without topology). Classes of Abelian groups that are closed under the 4 basic constructions are well-known: they coincide with varieties, i.e., with the classes of n-torsipn groups G (it means that nG = 0) for n € (jj. Those are classes of Abelian groups that are epireflective and monocoreflective at the same time. In [2], classes of Abelian groups that are reflective and coreflective at the same time are characterized as categories of modules over rings M that can be fully embedded into the category of Abelian groups; M is fully described - it is always countable and there are 2W of such non-isomorphic representations. A special case when M equals to the ring of rational numbers gives the class of divisible groups without torsion. We conjecture that there are no nontrivial reflective and coreflective classes in Gr. Now, if one takes the class C of all topological groups such their underlying group belongs to an epireflective and monocoreflective subclass of Ab, one gets an epireflective and coreflective class in TopAb. Unlike TopLin, Top and Unif, there are nontrivial coreflective classes of TopGr where the coreflection maps are not bimorphisms, even not monomorphisms. A class of topological groups is monocoreflective in TopGr iff it is closed under direct sums and quotients. It is bicoreflective iff, moreover, contains the group Z. Productivity of coreflective classes in TopGr was investigated in [7]. One difference in comparison with TopLin is also the fact that a /t-productive bicoreflective class in TopAb may contain some powers Gx for A > K and for some nontrivial G. As in Top and in TopLin it helps when we know a testing space for productivity: Proposition 6.1 (Herrlich and Husek in [7]) A bicoreflective class in TopGr is K-productive iff it contains ZA for X < K. Now it is not difficult to prove the following characterization of productivity numbers in TopGr. As in TopLin, all productivity numbers are infinite although in the present case finite products do not coincide with direct sums (but they are their quotients). Theorem 6.1 (Herrlich and Husek in [7]) The class of productivity numbers of bicoreflective classes in TopGr or in TopAb coincides with the class of submeasurable cardinals and {00}. The coreflective hull of all powers of Z is again a proper productive bicoreflective class in TopGr.

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If we take all monocoreflective classes into consideration, we are loosing the testing space for productivity and have similar situation as in Unif. Really, it has the same effect for productivity numbers: Theorem 6.2 (Herrlich and Husek in [7]) The class of productivity numbers of monocoreflective classes in TopGr or in TopAb coincides with the class of infinite regular cardinals and {00}. The procedure is similar to the first procedure in Unif described after Theorem 4.1. The corresponding class having productivity number equal to an uncountable regular cardinal K is the coreflective hulls of powers 5A, A < K, of a strongly rigid Abelian group S having cardinality at least K. Strongly rigid means here that S has trivial continuous endomorphisms only (see [18]). When we used LCS instead of TopLin we got measurable cardinals instead of submeasurable ones for productivity numbers. One can proceed similarly in TopGr and use a smaller surreflective category (e.g. of precompact Abelian groups) where we get also measurable cardinals for all productivity numbers. One needs that all the Abelian subgroups of topological groups under consideration are projectively generated by homomorphisms into topological Abelian groups having the property that for every sequence {xn} of nonzero elements there is a sequence {kn} of integers such that the sum Hknxn does not converge. 7. Summary

In the next table we summarize the results on productivity numbers for some categories and their possible connection with existence of testing spaces for productivity. In the last two columns, bi states for bicoreflective case and mono states for monocoreflective case. The abbreviations in the last raw should be clear (e.g., subm. stands for submeasurable). The trivial occurrences of 2 and of oo are missing in the last row. test space PC

Top

Topcfl

Unif

TopLin

LCS

TopGr bi

TopGr mono

2

2

-

R

subm.

reg.

2 subm.

-

subm.

R subm.

meas.

reg.

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References [1] Balcar, B. and M.Husek: Sequential continuity and submeasurable cardinals. Topfe Appl. Ill (2001), 49-58. [2] El Bashir, R., H. Herrlich and M. Husek: Abelian groups: Simultaneously reflective and coreflective subcategories. In: Categorical Perspectives (eds. J. Koslowski and A. Melton), Birkhauser 2001, 265-281. [3] Dierolf, P. and S.Dierolf: On linear topologies determined by a family of subsets of a topological vector space. Gen. Top. Appl. 8 (1978), 127-140. [4] Dow, A. and S.Watson: A subcategory of Top. Trans. Amer. Math. Soc. 337 (1993), 825-837. [5] Herrlich, H.: On the concept of reflections in general topology. In: Contributions to Extension theory of Topological Structures, Proc. Symp. Berlin 1967 (VEB, Berlin 1969), 105-114. [6] Herrlich, H.: Topologische Reflexionen und Coreflexionen. Springer Lecture Notes Math. 78 (1968). [7] Herrlich, H. and M. Husek: Productivity of coreflective subcategories of topological groups. Comment. Math. Univ. Carolinae 40 (1999), 551-560. [8] Husek, M.: Reflective and coreflective subcategories of UNIF and TOP. In: Seminar Uniform Spaces (ed. Z.Prolik), Prague 1973, 113-126. [9] Husek, M.: Products of uniform spaces. Czechoslovak Math. J. 29 (1979), 130-141. [10] Husek, M.: Special classes of compact spaces. Proc. of the Conf. on Categ. Top., Berlin 1978. Lecture Notes in Math. 719 (1979). [11] Husek, M.: Productivity of some classes of topological linear spaces. Top&App 80 (1997), 141-154. [12] Husek, M.: Productivity numbers in topological spaces, submitted [13] Husek, M. and M.D. Rice: Productivity of coreflective subcategories of uniform spaces. Gen. Top. Appl. 9 (1978), 295-306. [14] Kannan, V.: Reflexive cum coreflexive subcategories in topology. Math. Annalen 195 (1972), 168-174. [15] Mazur, S.: On continuous mappings on cartesian products. Fund. Math. 39 (1952), 229-238. [16] van Mill, J.: The Infinite-Dimensional Topology of Function Spaces. NorthHolland (Amsterdam) 2001. [17] Noble, N.: The continuity of functions on cartesian products. Trans. Amer. Math. Soc. 149 (1970), 187-198. [18] Shelah, S.: Infinite Abelian groups. Whitehead problem and some constructions. Israel J. Math. 18 (1974), 243-256. [19] Trnkova, V.: Non-constant continuous mappings of metric or compact Hausdorff spaces. Comment. Math. Univ. Carolinae 13 (1972), 283-295.

A CHARACTERIZATION OF CO-RETRACTS OF FUNCTIONAL STRUCTURES

ROLAND KASCHEK Department of Information Systems Massey University Palmerston North, New Zealand E-mail: [email protected] In the present paper we characterize the idempotent endomorphisms of structures. This characterization is used to characterize the retracts of structures in terms of certain equivalence relations on their support. For relation-free structures this result is rephrased to characterize co-retracts of structures in terms of wreath products of structures with admissible categories, a construction introduced in this paper. Mathematics Subject Classifications (2000): 08A05, 18B99 Keywords: structure theory, co-retraction, congruence, wreath product

1. Introduction Wreath products are a well-studied and useful construction, see Refs. [11, 7, 8]. The present paper generalizes the construction of wreath product of a monoid and a small category in Ref. [5]. The construction introduced here, i.e., the wreath product of a relation-free structure with a small and concrete category is intended to provide a mechanism for constructing new structures from given ones such that the new ones can be considered as a refinement of the old ones. Full detail concerning acts and monoids a topic only touched her can be found in Ref. [8]. In Applied Informatics large and complex systems are modelled and built. Respective development methodologies rely on what could be called step-wise model refinement. Many of the more frequently used models can be represented as structures. In some cases refinement can be modelled as obtaining co-retractions. The paper is supposed to lay the theoretical foundations for the construction of a computer based modelling environment for the more well-known and frequently used semantic models in Applied In157

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formatics such as the Entity-Relationship Model ([4]), Petri-Nets ([!]) or State Charts ([6]). Due to space restrictions examples and references about this are avoided. 2. Structures Definition 2.1. A vocabulary (T, a) is a pair where r = TF U TR is the disjoint union of a set TF of function symbols and a set TR of relation symbols and a : T —•» {0, 1, 2, 3, . . .} is a function associating to each symbol in r the number of its input parameters. A structure type z over a vocabulary (T, a) is a triple (T, aF, aR) where T is a finite set {71 , . . . , 7j} of so-called sorts and aF : TF —> { {1, . . . , /}|/ e TF} and aR : TR —> {tr : {1, . . . , a(r) + ! } — > { ! , . . . , l}\r € TR} are functions associating to the input or output parameters of symbols in T the sort they belong to. The elements a F ( f ) and aR(r) for f € TF and r £ TR often respectively are denoted as arity(f) and arity(r). The class of all structure types over vocabulary (T, a) is denoted by 9 TiQ . Let be z = (T,aF,aR) € 6T)Q then a quintuple S = (S,T,ts,Fs ,KS) is called z-structure or simply structure over (T, a), if the following assertions hold: S is a finite set, t : S —» T is a function and the sets i^ 1 (7i), are denoted as Si, Vi 6 {1, . . . , I}). Fs is a set {fs \ f £ TF, fs : x^Sju -» S,- artty(/) = is a set {rs |

r e rfl, rs C

In the sequel a vocabulary (T, a) will be denoted by its symbol set T only. According to the notation in [9] I denote the class of all z — structures over vocabulary T by (z) T or even by (z) if the vocabulary T is clear from the context or of no particular importance, Vz S QT. If S = (S, T, tg,J~s iH8) G (z), for a certain structure type z G 0 the set S1 is called the support of S and denoted by \S\. In case a structure is denoted by a calligraphic letter such as A, B or the like then the support \A\, \B\, ... is denoted by A, B, . . . respectively and further the structure is denoted by (A,T,t^, J:A,KA), (B, T, tB,FB , TIB), • • • respectively. Finally, if it is clear which is the support of structure S then the indices denoting the support are omitted. In this paper a structure is called functional if its vocabulary does not contain relation symbols.

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Definition 2.2. Let z = (T,aF,aR) £ 0 and A,B £ (z) then .4 is a substructure of B, if A C B, tA = tB\A, and fA = fB\A, rA = rB\A, V/ e TF and Vr € TR. To denote that .4 is a substructure of B it is written A B is called z-structure morphism, or just morphism, out of A into B if the following assertions hold: (1) tA = (2) 4>SA ^JB4>a(f\ V/ £ rF with ortty(/) = O'i, • • . , J a ( f ) J ) , (3) P(0)r^(oi,...,a a ( r ) )C r B 0 a W(ai,...,a a ( p ) ), Vr £ rfl with arity(r) = (ii,. . . ,i a ( r ),i) and afc e A ifc , Vfc e {1, . . . ,a(r)}.

The set of all z-morphisms out of .4 into 6 is denoted by HomZtT(A, B} or only as Hom(A, B) if the vocabulary or the structure type are assumed to be known or unimportant. If A = B then Hom(A,B) is denoted by End(A), its elements then are called z-endomorphisms or simply endomorphisms of A _ Instead of requiring in assertion 3 of definition 2.3 that P(4>)rA is a subset of rB(j)a^ from a categorical point of view one likely would request equality as it is done in [10]. With respect to the application of the results of the paper in Applied Informatics it appears that this could reduce the variety of the possible refinements of system models too much. Therefore only inclusion is requested here. Note that within this definition the following conventions are used (j> occurs x times (f>x = (j> x ... x W the function p(£) : P(V) -» P(W),X •-> U xex {£(z)} is introduced, where P(V) and P(W) are the power sets of V and W respectively. Note additionally that P(£C) = P(£)P((;), if the composition of £ and C is defined. Finally the convention rB : S^ x . . . x Si a(r) , (si,...,s a (r)) '-»{«€ 5j ( s i , . . . , s a ( r) ,Si) € r5} is used and similar for ,4 or other structures. In case there is no danger of confusion the lines over relation names as well as the powerset function operators are left out. Definition 2.4. Let r be a vocabulary and z = (T,aF,aR) £ 9, a structure type over T and A,B £ (z). Then <j> £ Homz(A,B) is called left invertible or co-retraction, if there exists ijj £ Homz(B,A), such that V = 1U, i-e. the identity on A, the support of A. Then V

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is called right invertible or retraction and A is called retract of B and the latter co-retract of the former. If a morphism is both left invertible and right invertible then it is called isomorphism. To denote that there is an isomorphism between structures A and B it is written A fa B. A morphism <j> e Homz(A,B) is called full, see, e.g. [2], if 4>(a) 6 rB(f)a^(ai,..., o a ( r )) C 4>(A) implies that the following assertions hold: (1) There (r)

exist

elements

a(,..., a',-.

€

A,

such

that

(r

(a) — <j)(a'), for all r e TR, with arity(r) = (ii,--.,ia(T),i) and au e Aiu, for all u e {!,..., a(r)}. A morphism e Homz(A, B) is called closed, see, e.g. [2], if r B (/> Q ( r ) (ai,... ,a a ( r )) C 0r A (ai,... ,a a(r )), for all r € TR, with arity(r) = (ii,...,ia(r),i) and au 6 A iu , Vu e {1,... ,a(r)}. Remark 2.1. a For each vocabulary r and each structure type z = (T, a F ,a fi ) £ 0 over r the following assertions hold true: (1) The composition of morphisms again is a morphism and the composition operation is associative. (2) For all S € (z) the mapping 1|S| is a morphism which is both left and right neutral to composition of morphisms. (3) (z) together with the z-morphisms is a category the morphism class of which is denoted by < z >. For the category concept see, e.g. [3]. (4) A z-morphism is an isomorphism iff it is bijective and full. (5) If a z-morphism is closed then it is full. (6) A bijective z-morphism is closed iff it is full. (7) Some full z-morphisms are not closed. Proof. The first three assertions are trivial. To prove that assertion 4 is true let at first (/> € Homz(A,B) be an isomorphism. Due to definition 2.4 then <j> is left- and right invertible and thus is bijective. It has to be shown that <j> is full. Assume for this end <j>(a) € rB(pa(-r^(ai,..., a a ( r )) C (A) for r € TH, with arity(r) = (ii,. •• ,ia(r),i) and au £ Aiu, for all a

The ideas how to define closed or full morphisms were developed according to [2]. The assertions 4,... ,7 are stated in the source. The example for the proof of assertion 7 however is a new one.

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u € {1,..., cn(r)}. Due to bijectivity of there is exactly the choice a = a' and afc = oj., V/c e {1,... ,a(r)}. Therefore it suffices to show that a' e r 4 ^,..., a^j,.))- Assume now that a' £ rA(a[,..., a'a,r)). Then the bijectivity of (/> implies that <j>(a'} £ (t>(rA(a(,... ,< (r) ))= P^r^oi,... ,< (r) ). Since ^ is a morphism the equation's right side term is a subset of r Thus the assumption implies a contradiction to r -B0«( )(a 1 ,. ,.,a a ( r )). the choice of <j)(a) and therefore must be false which establishes that is full. To show now that bijective full morphisms are isomorphisms let € Homz(A, B) bijective and full. It suffices to show that the inverse if) of <j) is a morphism. To see that assertion 1. of definition 2.3 holds true note that IA^— (tBet>)i}= *B- What had to be shown. To show that assertion 2. of definition 2.3 holds true let / € TF, with arity(f) = (ji,... , j a ( f ) , j ) Let further bk e Bjk, Vfc e {1,... ,a(/)}. Let further be ak = tp(bk}, Vfc 6 {!,..., a(/)}. Then V/ f l (&i, • • • , & « ( / ) ) = V/V ( / ) (ai, • • - , « « ( / ) ) • Since is a morphism this equals fAipa^\bi,... ,6 a (/)). Thus assertion 2. of definition 2.3 holds true and it is left to show that assertion 3. of this definition also holds true. To this end let r e TR, with arity(r) = (ii,... ,ia(r),i)- Let further be bk € Bjk,Vk 6 {!,. ..,a(r)}. It will be shown that P(ip)rB(bi,..., 6 a(r) )C ^V a(r) ( & i> • • • . 6 «(r))- Let thus a e P(^(bi,..., ba(r)). Then 0(a) e_P(4>)P(V>P~(li,..., & a (r,))= r s ^ a ^(oi,... o a ( r )). Since 0 is bijective r s is full, there are a(,... ,a',-. e A, such that 0(ofc) = (/>(a'k),Vk 6 {1,... ,a(r)} and <^(a) = 0(a'), for some o' 6 TA(o-'i,---,a-'a(r-\)- But this implies a = a' andafc = o^.,Vfc € {1,... ,a(r)}. Thus a e r^ai,..., a Q ( r ))= r-^V01^^!! • • • i&a(r))- So assertion 4. is proved. To show that assertion 5. holds let 0 e Homz(A,B) be a closed morphism. Let further r 6 TR, and 0,01,... ,o a (r) 6 ^4 such that <£(o) € r B ^ a ( r ) (oi,... ,a ai p/j a ( r ))C <^(yl). Thus r B (ai,...,oa(r))C ^r^(ai,... ,a a ( r )). This implies the existence of a' 6 r ^ ( o i , . . . ,a a ( r )), such that (a) = <j>(af). Since one can set a'k = Ofc,Vfc € {!,...,a(r)} the morphism 0 is full and assertion 5. is proved. To show that assertion 6. holds true it thus is sufficient to show that that bijective full morphisms are closed. Let thus 4> 6 Homz (A, B} be a bijective full morphism. Let further r e TR with arity(r) = (i\,.,., ia(r),i) and rB4>a(r\al,... ,o a ( r ) ) ± 0. Since (/> is bijective V6 € r B ^ a W(oi,... ,o Q ( r ) ), there is an a e A such that b = 0(o) € r B ^ a ^(oi,..., o a ( r ))C <^(A). Since (/> is full one has a 6 r ^ ( o i , . . . , a a ( r )) and so b = (/>(a) e ^r A (oi,... ,a Q ( r )), which implies that (j> is closed. To show that the last assertion holds true

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let r = TR = {r} with or(r) = 1, T = {Ti}, z = (T,9,aR) € 0T,a, with aR(r) =arity(r) = (1,1). Let further be .4 = ({a},T,i A ,0, {0}), and S = ({6,6'}, T, tB, 0, {{(6,6')}})- Compare the figure 1. Now let fj, : A —> B,a H-> b. Then //(a) ^ J"f/x(a) and so // is full. However /u is not closed. One finds rf (/u(a)) = {6'}, but /urf (a)= /u(0)= 0- And so rf (a) 2 ^V). D

Figure 1. A full but not closed structure morphism

Remark 2.2. Let r be a vocabulary, z = (T, aF, aR) e 0 a structure type over it and A, B 6 (2). Let further be ^ £ Homz(A, B). Then the following assertions hold: (1) If X < A, then the mapping <j>\x : \X\ —> \A\, is a morphism, the source restriction of <j> onto X. (2) If y y,a t-» (o) is a morphism, the target restriction of (f> onto 3^(3) Homomorphic images of structures are structures. Proof. Can be easily proven.

D

Definition 2.5. Let r be a vocabulary, z — (T,aF,aR) £ 6 a structure type over it and S = (S, T, t, F, K) 6 (z). A refinement c of ker(t) is called congruence, if the following assertions hold: (1) ( / s ( f l i , . . . , «„(/)),/Vi, • • - . * « ( / ) ) ) e c, V/ € TF with aritj/(/) = ( j i , - - - , J a ( / ) , j ) and SfeX e 5 jfc ,Vfc e {!,...,«(/)} with (s fc ,s' fe ) € c. (2) Vs € r j *(si,...,a a ( p ) )3a' 6 rA(s(,... ,a' ( r ) ), such that (s,s') e c, Vr e TR, with arity(r) = (ii,... ,ia(r),i) and Sk,s'k e 5jfc, with (s f c ,s' f c )ec,Vfc 6 {!,...,a(r)}.

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The set of equivalence relationships and the set of congruences on S respectively are denoted by equ(S), con(S). Further for an equivalence relation c on a set S the equivalence class of s G S often is denoted by [s]c or simply [s], if from the context is clear which equivalence the class is after. Remark 2.3. Let r be a vocabulary, z = (T,aF,aR) G 6, a structure type over it and S = (S,T,t,F,U) G (z) and c e con(S). Then S/c = (S/c,T,t/c,f/c,K/c) G (z), where: (1) t/c:S/c-*T,(s]~t(s), (2) TJC = {fs/c\ f G TF}, with fs/c : 5^/c x ... x S c o 0 ( f ) / -> 5j/c,([si],..., K(/)]) •-> [/(si,..., s a(/) )], if arity(f) = O'l,---,j'a(/),j)(3) 7£/c = {rs/c| r e rfi, with r s /c C 54l/c x ... x 5ia(r)/c x Sj/c, with arity(r) = ( i i , . . . ,i a ( r ) ,i) and sfc € Sit,Vfc £ {!,...a(r)}. The relation r s /c is denned such that ([si], • • • , [s a (r)]i [SD e r / c holds if and only if ( s i , . . . ,s a ( r ),s) € rs. Proof. It suffices to show that the relations and functions denned above are indeed well defined and that t/c is a function. Both are standard tasks and I omit the respective proofs. D Proposition 2.1. Let T be a vocabulary, z = (T,aF,aR) 6 6, a structure type over it and S € (z), c e con(S). Then the canonical mapping KC : S —> S/c, s H-» [s]c is a morphism onto S/c and its kernel ker(Kc) equals c. Proof. Since the kernel of a function is the set of pairs of its arguments that are identified by the function, i.e. are mapped onto the same image the proof can be easily obtained. D Proposition 2.2. Let r be a vocabulary, z = (T,aF,aR) e 6 a structure type over it, A,B € (z), and $ e Homz(A,B). Then fcer(^) G con(A). Proof. At first it will be shown, that ker(<j>) is a refinement of /cer(t^). Since 0 is a morphism IA = is^. Thus for (s, s') G ker() it holds *A(S)= *B(( S ))= tB(4>(s'))= tA(s'), and so (s,s') € ker(tA), what had to be shown. To show now that the functions of S are compatible with ker((/)}, i.e. that assertion 1. of definition 2.5 holds let / G TF, with a r i t y ( f ) = ( j i , - - - J a ( f ) J ) - Let further be sk,s'k G Sjk, with (sk,s'k) G fcer(0), Vfc G {!,..., Q(/)}. Then 0 Q <'>(si,...,s a ( / ) )= <^ ( / ) K> • • • > < ) - and so

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fB4>a(f\si, ..., s a (/))= / B since ^ is a morphism this equation implies 4>fA(s\, . . . ,s a (/))= 4>fA(s'i, • • • > S L(/))' Obviously this establishes the compatibility of functions with the equivalence relation. To show now that also the relations are compatible with ker(4>), i.e. that assertion 2. of definition 2.5 holds let K : A —> A/ her (). This mapping according to proposition 2.1 is a morphism K e Homz(A,B) and its kernel equalsfcer().Let now r 6 TR, with arity(r) = (ii, . . . ,i a ( r ) ,i). Let further be ak,a'k 6 Aik, with (afc,Oj.) e ker(),Vk e {1, . . . , a(r)}. Then « a < r >(ai, . . . , o o(r) )= «<*««, . . . , < (r) ). Hence (r' l /fc e r(0))K 0 'M(a 1 , • • • ,Oa(r))= (r- 4 /fcer()([a'1},. . . , [aa(r)]), such that [a] = [a']. Now, due to definition of rA/ker((j)} it follows [a'] e rA/ker()([a'1}, . . . , [o'a(r)])^ a' 6 rA(a(, . . . , a'atr\)- This proves the proposition. D Theorem 2.1. Let T be a vocabulary, z = (T,aF,aR) e Q, a structure type over it, S € (z), andc an equivalence relation on S. Then the following assertions are equivalent: (1) ce con(S). (2) 35' £ (z) and 4> £ Homz(S,S'), such that c = ker(<j>). Proof. Trivial with propositions 2.1 and 2.2.

D

Proposition 2.3. Let T be a vocabulary, z = (T,aF,aR) € Q a structure type over it and (f) e< z >. Then the following assertions hold: (1) If 4> is canonical then it .is full. (2) If (j) is a co-retraction then it is full. Proof. To show that the 1. assertion is true let A € (z] and let K : A —> A/c be the canonical mapping after the congruence c e con(A). Let r e TR, with arity(r) = (ii,..-,ia(r)^) arjd a i , . . . , o a ( r ) 6 A. Let (rA /c)Ka^(a\, . . . ,a a ( r ))C n(A). Let further K(O) 6 (r A /c)K a ( r ) (ai,. .. ,a a ( r )). Let a(,...,a'a^ € A, such that (ak,a'k) S ker(4>),Vk € {!,..., a(r)}. Since c is a congruence on A there 3a' e A, such that a' € rA/c(a'1,...,a'a,r'>), and [a] = [a']. Since c = ker(K) it follows K(O) = K(O'). Thus K is full. To show that the 2. assertion holdes let (j> : A —> S and V : S —* -4 such that

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ijj(j> = IA- Let r € TR with arity(r) = (i\, . . . ,i a (r)>*) and a/c € ^HI f°r au k € {!,..., a(r)}. Let further ^(o) G r«^ a W(oi,. . . ,o a(r) ) C 0(A). Then a = ^(a) G P(VO(r B tf a < r >(ai,...,a a ( r ) ))C ^ a(r V a(r) («i, • ••,<*««)= r yl (ai, . . . , a a ( r )). It thus is sufficient to chose at = a^., Vk G {1, . . . , o;(r)} to see that 4> ls fulld 3. Retracts of Structures Proposition 3.1. Let r be a vocabulary, z £ (T,aF,aR) G G astructure type over it and A, B G (2) . Then the following assertions are equivalent: (1) A is a retract of B. (2) There 3i2 = L G Endz(B), such that A « im(i). Proof. Assume at first that A is a retract of B. Then there are morphisms G Hom(A, B), ip G Hom(B,A), such that if>4> = IA- Define i : B —> B,b H-> (j)(il)(b)} . Due to remark 2.2 t is a morphism. The easy proof that £ : A -» im(i), a H-> ^(a), is a bijective morphism is omitted. It will be shown that the inverse £~ J : im(t) —> A of £ is a morphism. The image of b G im(i) under £-1 is the a G A such that £(a) = b. Consequently i[>(b) = ip((j)ip(j))(a) = a. Therefore £~ : = ip \im(i) and^" 1 is a morphism. Therefore assertion 1 implies assertion 2. To show that the reverse implication also holds true let L2 = L G Endz(B). Define (f> = IslimW and V> = t i m W . Then ^ and ip respectively are source restriction and target restriction of a morphism. Thus they are itself morphisms and it is easy to see that V>0 = l|im(<£)> which proves the proposition. D Proposition 3.2. Let T be a vocabulary, z = (T,aF,aR) G 6, a structure type over it, A,B,C G (z), <j) G Homz(A,B), surjective and if) € Homz(A,C) then the following assertions hold true: (1) (2) (3) (4)

ker((j>} C ker(if>), iff^-.B^C, such that £ = if). £ = ifxj)-1, if3£,:B^C, such that £<j> = if). ker(ip) = ker(), if 3^ : B —> C, such that^fi = tp and£ is injective. If 3£ : B —> C, such that (,<j> = ip, then tf> is surjective iff is surjective. (5) If(j) is full and3£, : B —> C, such that£4> = if>, then£ G Homz(B,C).

Proof. Since the result can be found in [2] only the 5. assertion needs to be proved since in the present paper the approach to full morphisms is

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somewhat different than in the cited source. Thus assume that is full and that 3£ : B —> C, such that ip = £0 holds. To show that ts = tc£ holds let b € B. Since ^ is surjective there is a & A such that 0(a) = b holds. Since (/>, ip are morphisms it follows that £5(6) = tB(a) = ^(a) = tcip(a). The assumptions thus imply that tci>(a) — tc£!, ...,6 a ( / ) )= VrV/ A (ai, ...,o a ( / ) ), where afc € -l(bk},Vk € {!,...,a(/)}. Now ^~V = ^er(0) C fccr(^), and so the right side term of the equation equals tpfA(ai, • • • i a a(/))i which, as one easily sees, is equal to fc(b\,... ,6 a (/)). This leaves to be shown, that also assertion 3. of definition 2.3 holds true. Let to this end r € TR, with arity(r) = (i\,. • • ,ia(r),i), and r° € *R.C'. Let further bk £ Bik,Vk € {!,...,a(r)}. Then P(Or F (6i,-.-,&a(r))= P(Or B 0 a ( r ) (oi,---,aa(r)), where afc_e ^(frfcJ.Vfc e {!,... ,a(r)}. Let now c € ^(O B (&i, • • • , &a(r))= P(£) rB 0 a(r) (ai> • • • ,aa(r))- Since (j> is surjective there exists <j>(a) €. rB(j)a^(ai,..., a a ( r )), such that c = £<j)(a). Since ^ is full and surjective there exist a',a(,... ,a',^ 6 A, such that a' £ rA(a'l,...,a'a(r}), <j>(a) = 0(a'), and 0(o fc ) =
Therefore c = ^(a')= ^(a')e PW^(ai,...,o^ ( r ) )C ...,6a(r)). which

is a morphism and so proves the proposition.

Therefore

shows that indeed £ D

Theorem 3.1. Let T be a vocabulary, z = (T,aF,aR) 6 O, a structure type over it, A,B& ( z ) , and jj, £ Homz(A, B}. Then there exists a factorization n = ifj.cffj.Kp., such that K^ is the canonical mapping after ker(n'), in is the identical inclusion of im((4) into B and the shadow a^ of n is a bijective morphism a^ : A/ker(n) —»zm(/u), [a] i-> /u(a). Proof. The canonical morphism due to 1. of proposition 2.3 is full. Since it also is surjective the assertions 1. and 5. of proposition 3.2 imply that there is £ e Homz(A/ker(iJ,), B), such that /i = £K M . It is easy to show, that the shadow of JJL is well defined and that im(£) = im(^). It is a standard matter to show that the shadow is a bijective morphism. Since obviously C — V7/' *ne theorem follows. D Corollary 3.1. Let r be a vocabulary, z = (T,aF,aR) e 6, a structure

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type over it, A £ (z), and fj, G Endz(A). Then the following assertions are equivalent: (1) M 2 = M > (2) {!,..., s},i H-> j, (2) V/ e TF with arity(f) 6~l(j)i such that

= (j,,. . . , ja(f), j)

(a) fA(xil,...,xtj) = x^^,...,^.), Vu e {!,..., a(f)},lu 0~ltiu),xiu €XiunA. (b) f

168

KASCHEK (3) Vr € aR, with arity(r) = (ii, ... ,ia(r),i)3pr 9~1(i), such that:

C xiu=10^l(iu) x

(a) xi e rA(xh,...,xlct(r)), iff I & pT(h, . . . ,la(r}), Vu e {!,..., a(r)}, Ju 6 fl-1^). (b) rA(Xh x ... x Xla(r)) C u,6M/1.....i n(r)) X,, V« e {!,..., a W K ^ e r - 1 ^ ) . Proof. Let A be a retract of B. Then due to proposition 3.1 there exists an idempotent /u2 = ^ € Endz(B), such that ,4 = im(n). According to corollary 3.1 this implies for c = fcer(/u), that there is a set {Xi, . . . , Xk} of congruence classes of B after c, such that Xj n A = {x^, Vi € {!,..., A;}, and /J-(Xi) C Xj. Therefore assertion 1 is seen to hold if one takes into account, that congruence classes always completely are contained in the instance set of a given sort. To show that assertion 2 holds true let / e aF with arity ( i i , - - . , i a ( f ) > i ) - To show, that the required f can be denned let ( l i , . . . , l a ( f ) ) e H"!^ s~l(Ju)- Then there are uniquely determined elements x\u £ XIH n A, Vu e {1, . . . , a ( f ) } , and IA(XII , . . . , xia(n ) & A. Thus there exists exactly one / S {1, . . . , fc}, such that fA(xit, ...,xij) = xi. Denote I with < / > f ( l i , . . . , l j ) . Thus <j>/ exists and assertion 2a has already been established. To show that 2b also holds true, let u € {1, . . . , «(/)}, /„ € G~l(ju). To assess the term fB(Xtl x . . .x Xia(n ), Iet6 u e X« u , V « e { t , . . . , a ( / ) } . Then (^.x/J e c, V« e {!,..., a(/)}. Since c is a congruence we have (/ B (Cij, • • • ,£,ii),fB(xil, . . . , x i - ) ) e c. Now fB(xil,...,xia(f)) =fA(xi1,...,xia(n) =o;^/(jl.....( Q ( / ) ) , and so e c. This implies / B ( & . • • • . & „ < / ) ) e BX ' >< ••• X •X'i holds and proves assertion 2b. To show now that also assertion 3 holds let r € TR with arity(r) = (ii, . . .,ia(r),i). Let (/i, . . . ,/ a ( r )) e !]"=! G~l(iu}. Then there are uniquely determined elements xiu £ Xiu n A, Vu e {!,.. .,a(r)}. Obviously ^(x^, . . . ,X( a ( r ) ) C A Thus there is a set pr(li, . . . ,la(r)) C {!,..., /e}, as was required to define the relation pr. By definition of pr assertion 3a holds true. To schow that also assertion 3b is true let lu e 0 l(iu), Vu 6 {1, . . . ,a(r)}. To investigate the term r B ( ^ ( l x . . . x ^ ( r ) ) , l e t a u € X I u , V « e {!,... ,a(r)}. Then (^ u ,xjj e c, Vu e {!,... i}- Since c is a congruence, for all £ e ^B(6! > • • • » &„<,.))» exists Z/ e r s (a;i 1 ,...,a;( a ( r ) ) =rA(xilt...,xia(r}), such that (£,xi) e c. Now xi e ^ ( ^ . . . . . X i ^ J . i f f Z e pr(k,...,la(r)). ThusrA(Xilx...xXia(r)) C p r ((j ..... /„(,.))• This establishes the necessity of assertions 1, 2, and 3. To show now that the conjunction of assertions 1, 2, and 3 is suf-

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ficient for A being a retract of B assume them to be true. Define H : B —> B,b i-> xi, if b 6 X;. Then obviously /i is idempotent and its image is A. It thus suffices to show that /u is a morphism. The definition of /x ensures that IB = tAH- Let / 6 rp , such that arity(f) = (h,---,ja(f),j)Let &„ e Xi u , and iu 6 fl" 1 ^) B Then M(/ (6i. • • • , & « < / > ) ) ^(fB(Xi, x . . . x X, a(/) ) But this set equals {^(d,...,;^,)} = {/ A (z«i>- • -,ZI Q { / ) )} = {A/^tei, • • •>$«<,)))}• Consequently M / B = /V (/) - Let now be r £ T£, such that arity(r) = (ii, . . . ,i a ( r ),i). Let again be £iu € X;u, and lu e fl-HO, V« e {1, . . . , a(r)}. Then Pd^fa , . . . ,6 a ( r ) ) C P^^Xi, x ...xX C P x U X, C U ( U / 6 p r ( i l...../ Q(r) This shows that in fact /x is a morphism and thus proves the theorem.

D

4. Wreath Products Definition 4.1. Let r be a relation-free vocabulary, i.e., TR = 0 and z = (T, aF, 0) e 6 a structure type over it. The set MT)2 of z-function terms is the intersection of all sets M on which a function a is defined such that the following properties hold : (1) r C M and a(p) = ((ji, . . . ,ja(P)), 0')). Vp e r with arity(p) = O'l)---.ja(p),j)(2) p o g e M, for all o-compatible p,q £ M, i.e., if o(p) = (inp,outp) and 0(9) = (in g , mp). In this case a(p ° q) = (ing, outp). (3) q\\r e M, for all |-compatible q,r € M, i.e., if 3p e M with a(p) = (in p , outp) and inp = outg||ouir, i.e., the concatenation of the tuples outq and outr. In this case a(^||r) = (inq\\inr,inp). Clearly the function a coincides on all sets M considered in the definition and is therefore inherited to M T(Z . The function a on it will be denoted as arity. Definition 4.2. Let r be a relation-free vocabulary, z = (T, aF, 0) a structure type over it and A e (z). A concrete and small category C is called ,4-admissible if the following assertions hold: (1) Va e A3Oa e O(C) such that a 6 Oa and Oa f~l O0< = 0 for a, a' e A with a ^ a'. (2) V j e {!,...

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KASCHEK (3) Vp € MTp0(3 = (j}p(f>g holds.

Remark 4.1. Let r be a relation-free vocabulary, z = (T, a F ,0) € © a structure type over it, A e (2) and C an ^-admissible category. Then MT:Z operates partially on O(C), i.e., the mapping A : MT,Z —> M(C), p i-> 0P has the property A(p o g) = A(p)A(g), for all o-compatible p, A^(C), a partial function with x ^ £ £ Homc(x,px), wherever px is defined and Homc(x,px} ^ 0} and the second component of which is is the partially defined composition (p, /) o (9i£0 = (P ° 9;/g3)i where fq is the partial function : O(C) —> .M(C), mapping x i-> f(qx), wherever 50; is defined and p, q are o-compatible and /gg is the composition of partial functions. The partial semigroup (W, o) here is called wreath product because of its multiplication that is typical for wreath products and because of the construction defined here is an obvious generalization of the wreath product of a monoid and a small category as defined in [5] if one concentrates on concrete categories. Remark 4.2. Let r be a relation-free vocabulary, z = (7", a F ,0) £ G a structure type over it, A G (z) and C an «4-admissible category. The composition o on W = AlC is associative where the respective compositions are defined, i.e., for (p, /), (q, g), (r, h) e W with o-compatible p, q and q, r we have ((p, /) o (q, g}) o (r, h) = (p, /) o ((q, g) o (r, h)). Proof. Let ( p , f ) , ( q , g ) , ( r , h ) 6 W with o-compatible p, q and q,r. Let RHS = (p,/) o ((q,g) o (r,/i)) = (p,/) o (q o r,grh). Now for x e O(C) for which rx is defined the morphisms (if they exist) g(rx) and h(x) can be composed. Thus RHS = (p, /)(
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all x 6 O(C) for which (q o r)x is denned and for which f ( ( q o j-)z) exists the morphisms f ( ( q o r)x) and g(rx)h(x) can be composed and Rff5 = (po(qor), f ( ( q o r)x)g(rx)h(x). Taking into account that p o (q o r) = (p o g) or where both sides are defined, i.e., when p, q and g, r are o-compatible and considering ((fqg)r)(x), which if all involved morphisms exist equals (fq9}(rx) = f(qrx)g(rx) results in (fqg)r = fqrgr and thus RHS = ((p, /)o ( fl)) ° (ri /O what had to be shown. D Theorem 4.1. Let T be a relation-free vocabulary, z = (T, a F ,0) e 0 a structure type over it and A€ (z). A z-structure B is a co-retract of A iff there exists an A-admissible category C such that the following assertions hold: (1) Bj=\Ja€AiOa,forallj€{l,...,\T\} and B = (Jje{1,...,\T\}Bj. (2) tB : B -> T, b ^ TJ, i f b € Bj. (3) pB o (TT-JJ x ... x 7Tj a(p) ) — <j>P, for all p 6 T with arity(p) =

Proof. Let at first be B be a co-retract of A. Then A is a retract of B. It needs to be shown that an ,4-admissible category C exists that fulfills the assertions 1, 2 and 3. According to theorem 3.2 there is an equivalence relation c on JB the classes of which have a singleton set intersection with A. For each a € A define Oa = [a\c. This implies Oa n Oa' = 0 for a, a' e A with a ^ o! . Use the disjoint union of sets and the identical inclusions as co-product family to define co-products. Use the cartesian product of sets as well as the projections of these sets onto their component sets and the cartesian product of functions to define products. Then assertions 1 and 2 of definition 4.2 are true. For p 6 T with arity(p) = (ji , . . . , ja(p) , j) define P • Il°=i BU -> BJ, (61, . . . , ba(p)) ^ pB(&!, . . . , 6 a(p) ). Then theorem 3.2 together with remark 4.2 implies that assertions 3 and 4 of definition 4.2 hold true. Thus an ,4-admissible category C can be defined. Clearly 1 and 2 hold true. That also the last assertions holds true follows directly from the definition of <j>p and of the definition of TT^, for all u & {1, . . . ,a(p)}. Thus the necessity of the condition has been shown. To establish the sufficiency of the condition let C be an ^.-admissible category. Let the assertions 1 to 3 hold true. It has to be shown that B as specified by these conditions is a structure over z and that it is a co-retract of A. That B indeed is a structure is a trivial consequence of the presupposition. To show that it is a co-retract of A let i : A —» -B, a H—> a and TT : B —> A, b H-* a, if b € Oa- Obviously in, — \A- It thus suffices to show that L and

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TT are morphisms. That i = \B \A is a morphism follows from remark 2.2. Since IB — IA^ holds by definition and the vocabulary is relation-free it suffices to show that assertion 2. of definition 2.3 holds true. It has to be shown that TtpB = pAna(p) holds for allp e r with arity(p) = (ji,... , j a ( P ) , J ) - Let for this be u € {1,..., a} and bu € Oaju C Bju. Then 7rp B (6i,..., 6 a(p) )€ 7r

P B ( x "=i ) Oa iu )C CV( 0jl ,..., aj . a(ij) ) = ^(aj,,...,^^).

equals p A 7r a ^(6i,..., 6 a ( p )) and the theorem is proven.

Obviously this D

References [1] Bernd Baumgarten. Petri-Netze. Bibliographisches Institut & F.A. Brockhaus AG, Darmstadt, 1990. [2] Peter Burmeister. A Model Theoretic Oriented Approach to Partial Algebras. http://www.mathematik.tu-darmstadt.de/ burmeister/, 2002. The book originally was published in 1986 by Akademie - Verlag, Berlin, GDR as volume 32 in the series "Mathematical Research". It now is available as a tex file at Peter Burmeisters home page at TU Darmstadt. 27.2.2002. [3] Michael Barr and Charles Wells. Category Theory for Computing Science. Prentice-Hall, New York et al., 1990. [4] Peter P. Chen. The Entity-Relationship Model: Toward a Unified View of Data. ACM Transactions on Database Systems, l(l):9-37, 1976. [5] Vladimir Fleischer and Ulrich Knauer. Wreath products of monoids with small categories whose principal one-sided ideals form tress. Journal of Algebra, 170(1):69 - 100, Nov. 1994. [6] David Harel and Amnon Naamad. The STATEMATE Semantics of Statecharts. ACM Transactions on Software Engineering and Methodology, 5(4):293-333, Oktober 1996. [7] Roland Kaschek. Uber das Endomorphismenmonoid des lexikographischen Produktes endlicher Graphen. PhD thesis, Fachbereich Mathematik der Universitat Oldenburg, 1990. [8] Mati Kilp, Ulrich Knauer, and Alexander V. Mikhalev. Monoids, Acts and Categories, volume 29 of de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin, New York, 2000. [9] Herbert Lugowski. Grundziige der universellen Algebra. BSB B.C. Teubner Verlagsgesellschaft, Leipzig, 1976. [10] J.J.M.M. Rutten. A calculus of transition systems (towards universal coalgebra). In Ponse, Alban and de Rijke, Maarten and Venema,Yde, editor, Modal logic and process algebra : a bisimulation perspective, pages 231 - 256. Center for the Study of Language and Information, Stanford, California, 1995. [11] Charles Wells. Some applications of the wreath product construction. American Mathematical Monthly, 83:317 - 338, 1976.

MAXIMAL (SEQUENTIALLY) COMPACT TOPOLOGIES

HANS-PETER A. KUNZI* Dept. Math. Appl. Math. University of Cape Town Rondebosch 7701 South Africa E-mail: [email protected] DOMINIC VAN DER ZYPEN Math. Institute University of Berne Sidlerstrasse 5 3012 Berne Switzerland E-mail: [email protected]

We revisit the known problem whether each compact topology is contained in a maximal compact topology and collect some partial answers to this question. For instance we show that each compact topology is contained in a compact topology in which convergent sequences have unique limits. We also answer a question of D.E. Cameron by showing that each sequentially compact topology is contained in a maximal sequentially compact topology. We finally observe that each sober compact TI-topology is contained in a maximal compact topology and that each sober compact Ti -topology which is locally compact or sequential is the infimum of a family of maximal compact topologies. Mathematics Subject Classifications (2000): 54A10, 54B15, 54D10, 54D30, 54G20 Keywords: maximal compact, KC-space, sober, US-space, locally compact, sequential, sequentially compact

* The first author acknowledges support under the bilateral cooperation between Flanders and South Africa (period 2003/4). 173

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1. Introduction A topological space is called a KC- space (compare also [5]) provided that each compact set is closed. A topological space is called a US-space provided that each convergent sequence has a unique limit. It is known [19] that each Hausdorff space (= Tg-space) is a KC-spa.ce, each KC-space is a US-space and each US-space is a Ti-space (that is, singletons are closed); and no converse implication holds, but each first-countable US-space is a Hausdorff space. A compact topology on a set X is called maximal compact provided that it is not strictly contained in a compact topology on X. It is known that a topological space is maximal compact if and only if it is a KC-space that is also compact [13]. These spaces will be called compact KC-spaces in the following. Let us note that while there are many maximal compact topologies, minimal noncompact topologies do not exist: Any noncompact space X possesses a strictly increasing open cover {Ca : a < 5} of X where 6 is a limit ordinal and Co can be assumed to be nonempty. Clearly then {0, X} U {Ca : 0 < a < 6} yields a base of a strictly coarser noncompact topology on X. Maximal compact topologies need not be Hausdorff topologies [17] (see also [1, 15]). A standard example of a maximal compact topology that is not a Hausdorff topology is given by the one-point-compactification of the set of rationals equipped with its usual topology. Indeed maximal compact spaces can be anti-Hausdorff (= irreducible), as we shall next observe by citing an example due to van Douwen (see [18]). In order to discuss that example we first recall some pertinent definitions. A nonempty subspace 5 of a topological space is called irreducible (see e.g. [6]) if each pair of nonempty open sets of S intersects. Furthermore a topological space X is called a Frechet space (see p. 53 [4]) provided that for every A C X and every x € A there exists a sequence of points of A converging to x. For the convenience of the reader we include a proof of the following observation (compare e.g. Math. Reviews 53#1519 [14]). Lemma 1.1. Each Frechet US-space X is a KC-space. Proof. Suppose that x £ K where K is a compact subspace of X. Because X is a Frechet space, there is a sequence (fc n ) n gN of points of K converging to x. Since K is compact, that sequence has a cluster point c in K. Because X is a Frechet space, there is a subsequence of (fc n )neN

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converging to c (compare Exercise 1.6D [4]). Hence x = c € K, because X is a US-space. We have shown that K is closed and conclude that X is a KC-space. Example 1.1. (van Douwen [18]) There exists a countably infinite compact Frechet US-space that is anti-Hausdorff. By the preceding lemma that space is a KC-space and hence maximal compact. Thus there exists an infinite maximal compact space that is irreducible. On the other hand, by the result cited above each first-countable maximal compact (Ti-)topology satisfies the HausdorfF condition (compare Theorem 8 [16]). 2. Main problem and related questions While it is known that each compact topology is contained in a compact TI-topology (just take the supremum of the given topology with the cofinite topology (see Theorem 10 [16])), the question whether each compact topology is contained in a compact KC-topology (that is, is contained in a maximal compact topology) seems still to be open. Apparently that question was first asked by Cameron on p. 56, Question 5-1 [3], but remained unanswered. Of course, a simple application of Zorn's Lemma cannot help us here, since a chain of compact topologies need not have a compact supremum: Consider the sequence (r n ) n6 N of topologies rn = {0,N} U {[l,fc] : k € N, k < n} (n € N) on the set N of positive integers. On the other hand, for instance each infinite topological space X with a point x possessing only cofinite neighborhoods is clearly contained in a maximal compact topology: Just consider the one-point-compactification Xx of X \ {x} where X \ {x} is equipped with the discrete topology and x acts as the point at infinity. The problem formulated above seems to be undecided even under additional strong conditions. Recall that a topological space is called locally compact provided that each of its points has a neighborhood base consisting of compact sets. Note that a locally compact KC-space is a regular Hausdorff space. Problem 2.1. Is each locally compact (resp. second-countable) compact topology contained in a maximal compact topology? The authors also do not know the answer to the following generalization of their main problem.

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Problem 2.2. Is each compact topology the continuous image of a maximal compact topology? In Example 11 [16] it is shown that a compact space need not be the continuous image of a compact T2-space. In fact, a careful analysis of the argument reveals the following general fact (also stated in 3.6 [7]). Proposition 2.1. A KC-space Y that is the continuous image of a compact T^-space X is a T^-space. Proof. Let / : X —> Y be a continuous map from a compact TVspace onto a KC-space. Clearly / is a closed map, since / is continuous, X is compact and Y is a KC-space. The conclusion follows, since obviously a closed continuous image of a compact T2-space, is a Ta-space. In this context also the following observation is of interest. Proposition 2.2. Let f : X —> Y be a continuous map from a maximal compact space onto a topological space Y. Then Y is maximal compact if and only if the map f is closed. Proof. Suppose that / : X —> Y is closed. Since f~l{y} is compact whenever y £ Y, we see that f~lK is compact whenever K is compact in y (compare e.g. with the proof of Theorem 3.7.2 [4]). Since f~1K is closed, we conclude that K = f ( f ~ l K ) is closed and hence y is a compact KC-space. For the converse, suppose that the map / : X —> Y is not closed. Consequently there is a closed set F in X such that fF is not closed. Clearly the compact set fF witnesses the fact that Y is not a KC-space. In connection with the preceding result we note (compare Example 3.2 [2]) that Ti-quotients of maximal compact spaces are not necessarily maximal compact. Problem 2.3. Are T\-quotient topologies of maximal compact topologies contained in maximal compact topologies? Next we want to show that a weak version of our main problem has a positive answer. Proposition 2.3. Let (X,r) be a compact T\-space. Then there, is a compact topology T' finer than T such that (X, T') is a US-space.

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Proof. As usual two subsets A and B of X will be called almost disjoint provided that their intersection is finite. Let M = {Ai : i 6 /} be a maximal (with respect to inclusion) family of pairwise almost disjoint injective sequences in X with a distinct r-limit (that is, each Ai € M is identified with {xn : n £ N) U {x} where (o;n)n6N is an injective sequence in (X,r) that converges to some point x different from each xn). For each i E / and m G N, let A™ = {xn : n e N, n > m} U {x}. Let T' be the topology on X which is generated by the subbase r U {X \ A^1 : i £ /, m £ N}. We first show that r' is compact. Let C be a subcollection of AT U {^4™ : i € /, n € N} with empty intersection. (Here, as in the following, AT denotes the set of r-closed sets.) Denote the intersection of C with AT by J-'. We want to show that there is a finite subcollection of C with an empty intersection. Of course, it will be sufficient to find a finite subcollection of C with finite intersection. If C — F, then such a finite subcollection of C must exist by compactness of (X, T). So in this case we are finished. If we have in our collection C \ f two sets Af and A™ with i ^ j, then their intersection will be finite. So in that case we are also done. Therefore we can assume that the set C\f is nonempty and its elements are all of the form A™ = {xn : n £ N, n > m} U {a} for some fixed io € / and n € M where M is a nonempty subset of N and a is the chosen r-limit of the sequence (a; n ) ne NIf a € n^7, then clearly a £ flC —a contradiction to flC = 0. So there is FQ e f such that a $. F0. Since FO is r-closed and the injective sequence (zn)neN r-converges to a, we conclude that F0 n {xn : n 6 N} is finite, since otherwise a £ F0. Hence for any m e M we have that F0 R A™ is finite and we are finished again. We deduce from Alexander's subbase theorem that the topology T' is compact. Next we want to show that (X, T') is a t/5-space. In order to reach a contradiction, suppose that there is some sequence (x n ) n6 N that r'-converges to x and y where x and y are distinct points in X. Replacing (x n ) n6 N if necessary by a subsequence, we can and do assume that the sequence (x n ) nS N under consideration is injective and that xn does not belong to {x, y} whenever n G N. The claim just made is an immediate consequence of the fact that the original sequence (a; n ) n6 N attains each value at most finitely many often, since (X, T) and thus (X, T') is a Ti-space and (x n ) n€ N has two distinct limits in (X,T'). Then (x n ) n6 N is an injective r-convergent sequence having a r-limit distinct from each xn and by maximality of the collection M there is some

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Ai = {zn : n e N} U {z} where z denotes the chosen r- limit of the sequence (zn)neN) belonging to M such that Ai n (xn : n € N} has infinitely many elements. Suppose that there is some p e N such that x or y does not belong to A?. Then X\A? is a r'-open neighborhood of x or y, respectively, which does not contain infinitely many terms of the sequence (x n ) n£ N which is impossible, because x and y are both r'-limits of (x n ) n6N . So there is no such p e N and it necessarily follows that x = z = y — a contradiction. We conclude that (X,r') is a US-space. Corollary 2.1. Each compact topology is contained in a compact UStopology. Remark 2.1. It is possible to strengthen the latter result further to the statement that each compact topology is contained in a compact topology with respect to which each compact countable set is closed. In order to see this we need the following two auxiliary results. We recall that a topological space is called sequentially compact provided that each of its sequences has a convergent subsequence. Lemma 2.1. Let X be a US-space and let {Kn : n £ N} be a countable family of sequentially compact sets in X having the finite intersection property. Then r\n^f^Kn is nonempty. Proof. For each n 6 N find xn € (~}f=lKi. Then the sequence (a;n)neN has a subsequence (yn)n£N converging to k & KI, because K\ is sequentially compact. Suppose that there is m € N such that k $ Km. Since there is a tail of (y n )n€N belonging to Km and Km is sequentially compact, there exists a subsequence of (y n )n€N converging to some p e Km. Since X is a US-space, it follows that k = p £ Km — a contradiction. We conclude that k e Lemma 2.2. Each compact US-topology is contained in a compact topology with respect to which each compact countable set is closed. Proof. Let (X, r] be a compact US-space and let T' be the topology generated by the subbase T U {X \K : K C X is countable and compact} ouX. We are going to show that T' is compact. In order to reach a contradiction, assume that C is a subcollection of Ar U {K C X : K is countable and compact} having the finite intersection property, but flC = 0. Since r is compact, we deduce that some compact countable set K belongs to C.

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Hence by countability of K there must exist a countable subcollection P of C such that HP = 0. Replace in V each member F of V D AT by its trace F f~l AT on K to get a countable collection V of compact countable sets having the finite intersection property. By a result of Levine [12], each compact countable space is sequentially compact and hence V is a countable collection of sequentially compact sets in a US-spa.ce. Since nP' = 0, we have reached a contradiction to the preceding lemma. We conclude that T' is compact by Alexander's subbase theorem. Evidently each compact countable set in (X, T') is r-compact and thus r'-closed. Problem 2.4. Given some fixed cardinal K > NQ. Is each compact topology contained in a compact topology with respect to which each compact set of cardinality K is closed? A modification of some of the arguments presented above allows us to answer positively the variant of the main problem (see Question 8-1 due to Cameron on p. 56 [3]) formulated for sequential compactness instead of compactness. Theorem 2.1. Each sequentially compact topology T on a set X is contained in a sequentially compact topology r" that is maximal among the sequential compact topologies on X. Proof. Since (X,r) is sequentially compact and any convergent (sub)sequence has a constant or an injective subsequence, it is obvious that any sequence in (X, T) has a subsequence that converges with respect to the supremum r Vr c where TC denotes the cofinite topology on X. Therefore by replacing r by r V rc if necessary, in the following we assume that the sequentially compact topology T on X is a Ti-topology. Define now a topology T' on X in exactly the same way as above (see Proposition 2.3). We next show that ( X , T ' ) is sequentially compact provided that (X,r) is sequentially compact. Let (y n ) n eN be any sequence in X. It has a subsequence (s n ) n eN that converges to some point a in (X,r), because (X, r) is sequentially compact. If (s n ) n eN has a constant subsequence, then (j/ n ) n gN clearly has a convergent subsequence in (X,T'). So by choosing an appropriate subsequence of (s n ) ne N if necessary, it suffices to consider the case that (s n ) n gN is injective and that sn ^ a whenever n € N. By maximality of M there is Ai = {zn : n 6 N} U {z} belonging to M. such that {sn : n €. N} n Ai is infinite. Hence there is a common injective subsequence of the injective sequences (sn)n^N and (-z n )neN in

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this intersection. By definition of r' that subsequence converges to z, because any basic r'-neighborhood G n n^=1(X \ A, j ) of 2; where n 6 N, G is r-open, Aj € M and kj € N (j = 1,... ,n) contains a tail of that subsequence, since (zn)neN r-converges to z and Aj n Ai is finite whenever j = 1,... ,n. We conclude that (y n ) n eN has a r'-convergent subsequence and that (X, T') is sequentially compact. As in the proof of Proposition 2.3 one argues that (X, T') is a US-space. We now define a new topology T" on X by declaring A C X to be r"closed if and only if xn e A whenever n e N and (z n ) n eN converges to x in (X, r') imply that x 6 A. It is well known and readily checked that T" is a topology finer than T' on X with the property that any sequence (o;ri)n6N that converges to x in (X,rr) also converges to x in (X, r"). In particular, it follows that the space (X, T") is sequentially compact, because (X, T') is sequentially compact. Let K be a sequentially compact subset in (X, T"). Suppose that xn € K whenever n € N and that the sequence (xn)neN converges to x in (X, T'). Then there is a subsequence (j/fc)fceN of (zn)«,€N that converges to r e K in (X, T'), since K is sequentially compact in (X, T") and T' C T". Thus x = r, since (Jf, r') is a US-space and hence x £ K. By the definition of the topology T" we conclude that K is closed in (X, T"). Therefore each sequentially compact subset of (X, T") is r"-closed. By Theorem 2.4 [2] we conclude that T" is a maximal sequentially compact topology on X, which is clearly finer than T. Let us finally mention another possibly even more challenging version of our main problem. Problem 2.5. Which (compact) Tj-topologies are the infimum of a family of maximal compact topologies? Evidently the cofinite topology on an infinite set X is the infimum of the family of maximal compact Hausdorff topologies of the one-pointcompactifications Xx (where x £ X) that we have defined above. In Proposition 4.1 below we shall deal with a special answer to Problem 2.5. 3. Some further results Let (X, T) be a compact topological space. Denote by AT (resp. CT) the set of all closed (resp. compact) sets of (X, T). Note that if r and r' are two compact topologies on a set X such that T C T', then AT C AT' C CT' Q C r - Of course, a topology r is a compact

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-ftTC-topology if and only if AT = CT. As usual, a collection of subsets of X that is closed under finite intersections and finite unions will be called a ring of sets on X. We consider the set MT of all rings Q of sets ordered by set-theoretic inclusion on the topological space (X, T) such that AT C Q C CT. Since AT is such a ring, MT is nonempty. If K, is a nonempty chain in MT, then (J/C belongs to .MT. By Zorn's lemma we conclude that MT has maximal elements. We shall call a collection C of subsets of a set X compact* provided that each subcollection of C having the finite intersection property has nonempty intersection. We use this nonstandard convention in order to avoid any confusion with the concept of a compact topology. Lemma 3.1. Let (X,r) be a compact topological space. If Q is a maximal element in MT that is a compact* collection, then Q = AT< where T' is a maximal compact topology finer than T. Proof. Suppose that Q is a maximal element in M.r that is compact*. Then {X \ K : K 6 Q} yields the subbase of a topology T' on X. Observe that AT C Q C AT' • Since Q is compact*, T' will be compact, by Alexander's subbase theorem. Because T' is compact, T C T' implies that ATi C CT. Hence AT> belongs to MT- We conclude that Q = AT' by the maximality oiQ. It remains to be seen that r' is maximal compact. If r" is a finer topology than T' and compact, then AT> C AT" C Cr. Hence by maximality of Q, AT" = Q = ATi and so r' = r". We have shown that T' is maximal compact. Proposition 3.1. Let (X, T) be a compact topological space such that each filterbase consisting of compact subsets has a nonempty intersection. Then T is contained in a maximal compact topology T'. Proof. Let Q be any maximal element in M.r as defined above. Recall that Q is closed under finite intersections. Hence any nonempty subcollection Q' of Q having the finite intersection property generates a filterbase consisting of compact sets on X. It follows from our hypothesis that Q is a compact* collection. Furthermore by Lemma 3.1 we conclude that Q is equal to the set of closed subsets of a maximal compact topology T' that is finer than T. It is known and easy to see (compare Theorem 6 [11]) that if X is a compact KC-space, then the product X2 is a KC-spa.ce if and only if X is

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a Hausdorff space. As an application of Proposition 3.1 we want to show however that the seemingly reasonable conjecture that the product topology of a large family of maximal compact topologies is no longer contained in a maximal compact topology is unfounded. In order to see this we next prove the following result. Lemma 3.2. Let (Xi)i&j be a nonempty family ofT\-spaces such that each Xi has the property that every filterbase of compact sets has a nonempty intersection. Then the product IL,6/Xj also has that property. Proof. We can (and do) assume that I is equal to some finite ordinal or an infinite limit ordinal e. Let f be a filterbase of compact subsets on the product IL1
n K)}.

It remains to show that for each Kef, {(y~f)~f<e €E TLy
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Proof. Note first that in a compact /fC'-topology each filterbase of compact sets has a nonempty intersection. We conclude by the preceding lemma and Proposition 3.1 that the compact product topology of an arbitrary nonempty family of maximal compact topologies is contained in a maximal compact topology. Corollary 3.1. Let (Xj)i 6 / be a nonempty family of spaces each of which is contained in a maximal compact topology. Then also their product topology is contained in a maximal compact topology. 4. Sobriety and maximal compactness Note that the closure of each irreducible subspace of a topological space is irreducible. Recall also that a topological space is called sober (see e.g. [6]) provided that every irreducible closed set is the closure of some unique singleton. Clearly each Hausdorff space is sober. Furthermore a subset of a topological space is called saturated provided that it is equal to the intersection of its open supersets. A short proof of the following result is given in [8]. Let {Ki : i G 7} be a filterbase of (nonempty) compact saturated subsets of a sober space X. Then |~)ie/ Ki is nonempty, compact, and saturated, too; and an open set U contains p|ie/ Ki iff U contains Ki for some i & I. Corollary 4.1. Let (X,r) be a compact sober Ti-space. Then T is contained in some maximal compact topology T'. Proof. Since all (compact) sets in a TI-space are saturated, the condition stated in Proposition 3.1 is satisfied by the result just cited. The statement then follows from Proposition 3.1. Problem 4.1. Characterize those sober compact topologies that are contained in a maximal compact topology. Remark 4.1. Let us observe that the maximal compact topology T' obtained in Corollary 4-1 will be sober, since the only irreducible sets with respect to the coarser topology r are the singletons. Van Douwen's example [18] mentioned earlier shows that a maximal compact topology need not be (contained in) a compact sober topology. Example 4.1. Note that the closed irreducible subsets of the one-pointcompactification X (of the Hausdorff space) of the rationals are the singletons: Any finite subset of a TI -space with at least two points is discrete

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and hence not irreducible. Moreover any infinite subset of X contains two distinct rationals and thus cannot be irreducible. We conclude that an arbitrary power of X is a compact, sober T\ -space, because products of sober spaces are sober (see e.g. Theorem 1.4 [6]). In the light of the proof of Proposition 3.1 one wonders which compact sober T\ -topologies can be represented as the infimum of a family of maximal compact topologies (compare Problem 2.5). Our next result provides a partial answer to this question. An interesting space satisfying the hypothesis of Proposition 4.1 is a TI-space constructed in [10]: It has infinitely many isolated points although each open set is the intersection of two compact open sets. (It was noted in the discussion on p. 212 [10] that that space is locally compact and sober.) Recall that a topological space X is called sequential (see p. 53 [4]) provided that a set A C X is closed if and only if together with any sequence it contains all its limits in X. Proposition 4.1. Each compact sober TI -space (X, T) which is locally compact or sequential is the infimum of a family of maximal compact topologies. Proof. Note that if K belongs to the closed sets of a maximal compact topology a finer than T, then K is compact with respect to a and thus with respect to r. In order to verify the statement, it therefore suffices to construct for any compact set C that is not closed in (X, T) a maximal compact topology a finer than r in which C is not closed. So let C be a compact set that is not closed in (X, T). In (X, T) we shall next find a compact set KQ such that KQ n C is not compact. Suppose first that X is locally compact. Then there is x £ X such that x € clTC \ C. Let F = {K : K is a compact neighborhood at x in (X, T)}. Of course, nF = {x}, since X is a locally compact Tj-space. Suppose that K n C is compact in (X, T) whenever K € f. Then {K n C : K G J-} is a filterbase of compact saturated sets in X. According to the result cited above from [8], we have nFnC — f\{Kr\C: K e T} ^ 0. Since x £ C, we have reached a contradiction. Thus there is a compact neighborhood K0 of x such that K0 n C is not compact in (X, T). Suppose next that X is sequential. Since C is not closed, there is a sequence (xn)neN in C converging to some point x e X such that x does not belong to C. Assume that {({x} U {xn : n e N, n > m}) n C : m e N} is a filterbase of compact sets. Clearly its intersection is empty, because

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r is a Ti-topology and (x n ) ne N converges to x—a contradiction. Hence there is m e N such that ({x} U {xn : n 6 N, n > m}) n C is not compact. Denote the compact set {x} U {xn : n £ N, n > m} by KQ. So our claim holds in either case. Note now that T U {X \ KO} is a subbase for a compact topology r' on X that is also sober and T\. By Corollary 4.1 there is a maximal compact topology r" finer than T'. Observe that X \ C £ T" : Otherwise C £ AT» and, since KQ € A-", also K0 n C £ AT». Therefore K0 n C € Cr» and .Ko n C e Cr —a contradiction. Thus indeed X \ C (£ T". We conclude that r is the infimum of a family of maximal compact topologies. Observe that the argument above also yields the following results. Corollary 4.2. Each locally compact sober T\-space in which the intersection of any two compact sets is compact is a KC-space (and therefore is a regular Hausdorff space). Corollary 4.3. Each sequential sober Ti-space in which the intersection of any two compact sets is compact is a KC-space. We next give an example of a compact sober Ti-topology that is not the infimum of a family of maximal compact topologies. Example 4.2. Let Y be an uncountable set and let — oo and oo be two distinct points not in Y. Set X = Yu{—oo, oo}. Each point ofY is supposed to be isolated. The neighborhoods of oo are the cofinite sets containing oo and the neighborhoods of — oo are the. cocountable sets containing —oo. Clearly X is a compact sober T\ -space. Next we show that with respect to the defined topology T a subset A of X is compact and not closed if and only if A is uncountable, oo € A and — oo ^ A : Indeed, if oo £ A, then A is clearly compact and if A is uncountable and — oo $ A, then A cannot be closed. In order to prove the converse suppose that A is compact and not closed in (X,r). Then A is certainly infinite. It therefore follows from compactness of A that oo e A. Since A is not closed, we conclude that —oo e A \ A and hence A is uncountable. Of course, if T' is a maximal compact topology such that T C T', then AT C AT' C CT' C Cr. Observe that the topology T" generated by the subbase {{—oo}} U T clearly yields a compact T-2-topology finer than T. Obviously, CT \ AT C AT" by the description found above of the nonclosed compact sets

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in (X, T). Thus AT" = CT. We conclude that T" is finer than any maximal compact topology containing T. Hence T" is the only maximal compact topology (strictly) finer than T. Let us recall that a topological space is called strongly sober provided that the set of limits of each ultrafilter is equal to the closure of some unique singleton. Of course, each compact Hausdorff space satisfies this condition. We finally observe that each locally compact strongly sober topological space (X, r) possesses a finer compact Hausdorff topology; just take the supremum of T and its dual topology (see e.g. Theorem 4.11 [9]). By definition, the latter topology is generated by the subbase {X \ K : K is compact and saturated in X} on X. No characterization seems to be known of those topologies that possess a finer compact Hausdorff topology. References [1] V.K. Balachandran, Minimal bicompact space, J. Ind. Math. Soc. (N.S.) 12 (1948), 47-48. [2] D.E. Cameron, Maximal and minimal topologies, Trans. Amer. Math. Soc. 160 (1971), 229-248. [3] D.E. Cameron, A survey of maximal topological spaces, Topology Proc. 2 (1977), 11-60. [4] R. Engelking, General Topology, Heldermann, Berlin, 1989. [5] W.G. Fleissner, A Tg-space which is not Katetov TB, Rocky Mountain J. Math. 10 (1980), 661-663. [6] R.-E. Hoffmann, On the sobrification remainder sX — X, Pacific J. Math. 83 (1979), 145-156. [7] R.-E. Hoffmann, On weak Hausdorff spaces, Arch. Math. (Basel) 32 (1979), 487-504. [8] K. Keimel and J. Paseka, A direct proof of the Hofmann-Mislove theorem, Proc. Amer. Math. Soc. 120 (1994), 301-303. [9] R. Kopperman, Asymmetry and duality in topology, Topology Appl. 66 (1995), 1-39. [10] H.-P.A. Kiinzi and S. Watson, A nontrivial T\-space admitting a unique quasi-proximity, Glasgow Math. J. 38 (1996), 207-213. [11] N. Levine, When are compact and closed equivalent?, Amer. Math. Monthly 72 (1965), 41-44. [12] N. Levine, On compactness and sequential compactness, Proc. Amer. Math. Soc. 54 (1976), 401-402. [13] A. Ramanathan, Minimal-bicompact spaces, J. Ind. Math. Soc. (N.S.) 12 (1948), 40-46. [14] M.J. Reed, Hausdorff-like separation properties and generalizations of the first countability axiom, Tamkang J. Math. 5 (1974), 197-201.

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[15] N. Smythe and C.A. Wilkins, Minimal Hausdorff and maximal compact spaces, J. Austral. Math. Soc. 3 (1963), 167-171. [16] A.H. Stone, Compact and compact Hausdorff, in: Aspects of Topology, pp. 315-324, London Math. Soc., Lecture Note Ser. 93, Cambridge Univ. Press, Cambridge, 1985. [17] H. Tong, Note on minimal bicompact spaces (preliminary report), Bull. Amer. Math. Soc. 54 (1948), 478-479. [18] E.K. van Douwen, An anti-Hausdorff Frechet space in which convergent sequences have unique limits, Topology Appl. 51 (1993), 147-158. [19] A. Wilansky, Between 7\ and T2, Amer. Math. Monthly 74 (1967), 261266.

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SUPERTOPOLOGIES AS STARTING POINTS FOR GENERALIZED CONTINUITY STRUCTURES

D. LESEBERG Department of Mathematics and Informatics Free University of Berlin E-mail: [email protected] Two interesting generalizations of supertopologies in the sense of Doitchinov are considered. The first one of these leads us to generalized nearness spaces, and consequently topological extensions are being examined. Secondly we obtain equiconvergence spaces which fill the gap between uniform convergence spaces and the non-symmetrical Kent-convergence spaces resulting as a convenient topological category which is cartesian closed and extensional. Mathematics Subject Classifications (2000): 54A05,54A20,54C20,54C35, 54D35,54E05,54E15,54E17 Keywords: topological constructs, posets, cartesian closedness, compact extensions, supernearness, filterisotopies, convergence, equiconvergence.

1. Introduction Supertopologies in the sense of Doitchinov [8] are being described by axiomatizing surrounding niters of sets. In special cases we obtain back the notion of arbitrary topologies and proximities respectively. By considering these niters or their corresponding cross-products as generating units we are able to formulate more general structures such as supernear operators or filterisotopies respectively which leads us in a special case to the so called equiconvergence structures. In the first case we obtain a generalization of the nearness structures in the sense of Herrlich [11], and it is possible to characterize those supernear spaces which can be extended to topological ones. A one-to-one correspondence between compact extensions and contigual supernearness is established. Secondly we also have a generalization of uniform convergence spaces in the sense of [21], and especially equiconvergence is being considered as a special kind of these so called filterisotopies, i.e., we axioma189

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tize uniform filters converging to suitable points. Thus, simple convergence as well as uniform convergence bothly can be described in such a manner. Moreover, it turns out that the corresponding category is set-like enough to allow constructions of special objects such as function spaces and universal one-point extensions and thus denning a topological universe. 2. Super near Spaces As usual, PX denotes the power set of X, and we use Bx to denote a collection of bounded subsets of X, also known as a B-set, i.e., Bx satisfies the following three axioms: (Bl) B' C B e Bx implies B' e Bx; (B2) 0 e Bx; (B3) x e X implies {x} € Bx. If Bx and BY are B-sets on X and Y, respectively, a function / : X —> Y is called bounded, if it preserves bounded sets. BOUND denotes the category whose objects are B-sets and whose morphisms are the bounded maps. We note that BOUND is a topological universe, which means it is a topological category having the fine properties of being cartesian closed and extensional. We recall the confinement relation P(P(PX}) is called a supernear operator or a supernearness on Bx, and the pair (Bx, 5) is called a supernear(ness) space, iff (SN1) (SN2) (SN3) (SN4) (SN5) (SN6)

B € Bx and Si < S2 e S(B) imply Si € 5(B); 5(0) = {0} and Bx £ S(B) for each B e Bx; B' C B e Bx implies S(B') C 5(5); a; € X implies {{x}} e 5({x}); B e S^ and Si U «S2 e 5(B) imply 5i 6 5(B) or S2 e 5(B); B 6 Bx and (cls(F) | F e 5} £ 5(B) for some 5 C P(PX) imply 5 e 5(5), where ds(F) := {x 6 X {{x}, F} e S({x})}.

Elements of S(B) are called B-near collections. Given a pair of supernear spaces (Bx, 5), (BY, T), a bounded map / : X —> Y is called a supernear map or shortly sn-map, iff

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(SN) B e Bx and S e 5(5) imply {/[F] F € 5} e T(/[5]). A map will also be referred to as a supernear map by saying it preserves B-near collections in the above sense. We denote by SN the corresponding category. Example 2.1. (1) Now let there be given a supertopological space (X, 9), that means, 0 is a map from the underlying B-set Bx to the set of all niters on X such that:

(ST1) 9(0) = PX; (ST2) B £ Bx and U e 0(5) imply U 2 B(ST3) B' CB&BX imply 9(5) C 9(5'); (ST4) B £ Bx and [7 e 9(5) imply there exists a set I/ € 9(5) such that always U £ 9(5') for each B' & Bx with B' C V. We obtain a supernear operator on Bx by setting 50(5) := {S C PX S C sec9(5)}, where sec9(5) = {A & PX W e 9(5). A n C/ / 0}. We also note that 0(5) denotes the neighbourhood system of B with respect to 9. sec9(5) is then a system of B-near collections, and we consider all subcollections of this type. Conversely, let there be given a supernear operator T, then an underlying neighbourhood-operator QT can be constructed as follows: 9T(5) := {U C X | V5 £ S(B). U 6 sec«S}. That special case thus leads us to an one-to-one correspondence between all supertopologies on a B-set Bx and the so defined generated supernear operators on it. Consequently we obtain a full embedding of the category STOP to SN. (2) Consider a B-set Bx on X. For a nearness structure £ on X we obtain a supernear operator on Bx by setting:

f{0}, Sf(B) := \ ( {S C PX

if 5 = 0

SU {B} e £}, otherwise

(3) For a Kuratowski closure operator cl(—) on X, we obtain a supernear operator on Bx by setting: 5ci(B) := {S C PX | B e sec{cl(-F) | F e S}}.

192

LESEBERG (4) For a Leader-proximity [18] 6 on X we obtain a supernear operator on Bx by setting: SS(B) :={SCpX

SC 6(B)},

where S(B) := {F C X \ B 6 F}. (5) For a quasi-uniformity U on X [7] we obtain a supernear operator on Bx by setting: SU(B) := {S C PX | VU € W. f|{C/(F) | F e 5 U {B}} ^ 0}, where Z7(F) := {?/ 6 X | 3x £ F. (x, y] € U}. (6) Doitchinov introduced the notion of supertopological spaces in order to construct a unified theory of topological, proximity and uniform spaces, and he proved a certain relationship of some special classes of supertopologies — called 6-supertopologies — with compactly generated extensions. By a compactly generated extension we understand a triple E := (e, Bx , Y), where X = (X, clx), Y = (Y, cly) are topological spaces (given by closure operators), Bx is a B-set on X and e : X —> Y is a function satisfying the following conditions: (CE1) A e PX implies dx(A) = e-1[dY(e[A])]; (CE2) cly(e[A|) = Y, which means that the image of X under e is dense in Y; (CE3) x 6 X and y e cly({e(z)}) imply e(x) e cly({j/}), which means that Y is symmetric relative to e[X}. (CE4) {cly(e[J4]) | A C X} is a base for the closed subsets of Y, which means that the extension E is strict in the sense of Banaschewski [2], (CE5) For any y e Y there exists a set A C X such that y € cly(e[j4]), and cly(e[.A]) is compact, which means that the extension is compactly generated. Now, CEXT denotes the category where objects are the just defined triples E = (e,Bx,Y). Morphisms in CEXT have the form (f,g) : (e,Bx,Y) —> (e',Bx',Y'), where / : X —-» X', g : Y — > Y' are continuous maps such that / is also bounded, and the following diagram

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commutes:

x --—-y 'I !• I ) x1 --—^ y If (f,g)t: (e,Bx,Y) -^ (e',Bx',Y'} and (/',«,') : (e',Bx',Y') -+ (e", Bx> , Y") are CEXT-morphisms, then they can be composed according to the rule (/', g') o (/, g) = (/' °f,g'og): (e, BX,Y) —> (e" , Bx , Y"), where o denotes the composition of maps. Given a compactly generated extension E = (e, Bx , Y), we now obtain a supernear operator on Bx by setting: SE(B) := {S CPX

VF e Sly e cly(e[B]).y e cly(e[F])}.

Remark 2.1. Before studying this relationship between CEXT and SN we pointed out that — in correspondence to the above-mentioned examples — the category SN of supernear spaces contains the following categories as full subcategories: - the category TOP of topological spaces and continuous maps; - the category PROXLE of Leader proximity spaces and <J-maps, hence also PROXLOi the category whose objects are Lodato proximity spaces; - the category NEAR of nearness spaces and nearness-preserving maps; - the category CONT of contiguity spaces and c-maps; - the category UNIF of uniform spaces and uniformly continuous maps. Lemma 2.1. For a compactly generated extension E = (e,Bx ,Y) the supernear operator SE has the following additional properties: (S) SE is symmetric, which means B&BX andSeSE(B) imply {B}(JS€ r\{SE(F) I F € (S n Bx) U {B}}. E (A) S is additive, which means B1UB2& Bx implies SE(Bl U JB2) C SE(Bl) U SE (52). E (Cl) S is closure-isotone, which means c\SE(B) e Bx implies SE(dSE(B)) C SE(B).

194

LESEBERG (E) SE is endogenous, which means B&BX implies \J{S C PX \ S e SE(B)} € SE(B).

Moreover, the closure operator clgE concides with the topological closure operator c\x • 3. Functorial Relationship between CEXT and SN Theorem 3.1. We obtain a functor F : CEXT —> SN by setting (a) F(E) := (BX,SE); for a compactly generated extension E = (e,Bx,Y) (b) F ( f , g ) := f for a CEXT'-morphism (f,g) : (e,Bx,Y) — * (e',Bx',Y'). Proof. In view of Lemma 2.1 we already know that F(E) is an object of SN with the corresponding additional properties. Now, let (f,g) : E = (e,Bx,Y) —> E' = (e',Bx',Y') be a CEXTmorphism. It has to be shown that / preserves the near- collections from F(E) = (BX,SE) to F(E') = (BX',SE'). Without loss of generality, let B 6 Bx \ {0} and S 6 SE(B). Now consider F € S. By definition, there exists y £ cly(e[5]) such that y € cly(e[F]). The hypothesis implies g(y) 6 g[c\Y(e(B})\ and therefore g(y) e clY>(g[e[B]]) = c\Y>(e'[f[B]]), since (/, g) is a CEXT-morphism. Because y € cly(e[F]), we have g(y) e dY<(e'[f[F}}), which results in {/[F] | F e S} e SE' (f[B}). D To obtain a related functor in the opposite direction, we introduce the notion of so-called B-clips for each bounded set B € Bx \ {0}. This is motivated by the following facts. Given a (compactly generated) extension E = (e,Bx,Y), it is possible to define a function t : Y —> P(PX) by setting: t(y) :={TCX\y€ cly(e[T])}. Moreover, for each B € Bx \ {0} we put: CB:=\J{t(y)\y£c\Y(e[B})}. Now, every B-near collection S £ SE(B) satisfies S C CB; in fact F € S implies the existence of some y € cly(e[J3]) such that y G cly(e[jFj), hence F £ t(y) and consequently F 6 CB . This leads to the following definition.

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Definition 3.1. Let (BX,S) be a supernear space. For B £ Bx \ {0} a subset C C PX is called a B-clip in S, provided that

(Cl) (C2) (C3) (C4) (C5) (C6) (C7)

0£C; Ci 6 C and Ci C C2 € 7>X imply C2 6 C; Ci U C2 e C implies Ci e C or C2 e C; 5 e C; cls(C) e C implies C e C; C e 5(B); n{clsCn I T1 e C} = 0 implies the existence of a finite subset C0 C C with fKclsCO | T € C0} = 0.

Another interesting example for this notion is given by the set system ex(x) := {T C X \ x 6 cls(T)} for x £ X, which is a {i}-clip in S. Moreover, ex(x) is a maximal element in S({x}) ordered by set inclusion. With respect to the above-mentioned motivation and remarks, we naturally arrive at the following definition. Definition 3.2. A supernear space (BX,S), as well as S, is called clipdetermined, provided that (Cl) B e Bx \ {0} and S € S(B) imply the existence of a B-clip C with SCC. Remark 3.1. In addition to the properties of Lemma 2.1, the supernearness SE as defined in Example 2.1(6) is also clip-determined. In preparing the introduction of a functor G : SN —> CEXT in the opposite direction to F we note the following: for a supernear space (Bx , S) we put: X := {C e PX | 3B e Bx \ {0}.C is a B-clip} and for each A C X we set c\x(A) ~ {C e X

f}AcC},

where QA := {F C X | VC € A-F e C} (so that, by convention, f|-4 = PX, if ^4 = 0). Then cl^ is a topological closure operator on X, which is easy to verify. Theorem 3.2. For supernear spaces (Bx ,S) and (BY ,T) let f : X —> Y be an sn-map. Define a function f : X —> Y by setting for each C e X f(C):={DCY\f-l(c\T(D)}€C}.

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Then the following statements are valid: (1) f is a continuous map from (X,clg) to (y,clp). (2) The composites f o e\ and ey ° f coincide, where ex '• X —> X is the function that assigns the {x}-clip ex(x) to x. (3) {f[C\ C e C } C / ( C ) . (4) Rex [B] := n{ex(x) | x £ B} = {F C X Be ds(F)} for every BCX. Proof. This can be shown in a straightforward manner.

D

Remark 3.2. With respect to Lemma 2.1 and Remark 3.1 we summarize that the supernear operator SE satisfies the axioms of being symmetric, additive, closure-isotone, endogenous and clip-determined. These facts motivate the following notion. Definition 3.3. A supernear operator on Bx, and also the corresponding space, is called contigual, if the above-mentioned axioms for the operator are satisfied. Moreover, we denote the corresponding full subcategory of SN by CSN. Theorem 3.3. We obtain a functor G : CSN —> CEXT by setting (a) G(BX ,S) := (ex, Bx, X) for any contigual supernear space (Bx, S) with X = (A", cls) and (X,cLj); (b) G(f) := (/,/) for any sn-map f : (BX,S) —> (BY,T). Proof. In view of (SN6) it is straightforward to verify that els is a topological closure operator on X. By Remark 3.1, we also have the topological closure operator clj^ on X. Therefore we obtain topological spaces with the B-set Bx, and ex '• X —> X is a continuous map according to Theorem 3.2. The proof of (CE1) till (CE3) is left to the reader. Axiom (CE4) can be verified in an indirect manner, and (CE5) should be proven according to (C7) in the definition of a B-clip in S. D Theorem 3.4. Let F : CEXT —> SN and G : CSN —> CEXT be the functors given in Theorems 3.1 and 3.3. For each object (BX,S) o/CSN let t(Bx ,S) denote the identity map t(Bx,S) := idx : F(G(BX,S)) —» (Bx, S).

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Then t : FoG —> ICSN is a natural isomorphism of FoG and the identity functor ICSN, «-e., idx : F(G(BX,S}) —> (Bx, S) is an isomorphism for each CSN-object (BX,S), and the following diagram commutes for each sn-map f:(Bx,S)—+ (BY,T): F(G(BX,S))

—

*- (Bx, 5)

F(G(BY,T))Proof. The commutativity of the diagram is obvious, since F ( G ( f ) ) = f . It remains to prove that idx : F(G(BX,S)) —> (BX,S) is an sn-map for each object (Bx, S) of CSN and vice versa. But to show this is left to the reader. D Now we are able to formulate the main theorem, which is a consequence of the preceding Lemmata and Theorems, respectively. Theorem 3.5. Let (BX,S) be a supernear space. Then the following are equivalent: (1) (BX,S) is contigual; (2) there exists a compactly generated extension (e,Bx,Y) such that for each B € Bx \ {0} the elements S € S(B) are characterized by cly(e[B]) 6 sec{cly(e[F]) F € S}; (3) there exists a topological space (Y, cly) and a continuous map f : X —> Y that satisfies -

cls(A) = /-1[cly(/[A])] for each A C X; f[X] is dense in Y; Y is symmetric relative to f[X]; {cly(e[Aj) | A C X} forms a base for the closed subsetes ofY; Vy e Y3A C X. y e cly(e[A]) and cly(e[^4]) is compact; for each <5 6 Bx \ {0} the elements S € S(B) are characterized by the fact that for each F & S there exists y € cly (e[Bj) such thaty£c\Y(e[F}).

4. Filterisotopic Spaces It has been a trend in Categorical Topology to embed topological categories into topological universes, i.e., topological categories which are set-

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like enough to allow constructions of special objects, such as function spaces or universal one-point extensions. The topological universes of preuniform convergence spaces and semiuniform convergence spaces introduced by Preufi [19] form together a common generalization of symmetric limit spaces (and thus also symmetric topological spaces) and uniform limit spaces (and thus also uniform spaces). Moreover, these universes have the additional property that products of quotients are again quotients, hence they are also strong. The above pictured list summarizes the connections between uniform and topological convergence, morover these facts are being bridged over the filtermerotopic spaces in the sense of Katetov [15] or the Cauchy spaces respectively. PUCONV

SUCONV

GConv

SULim

KConv

ULim

KConv,

Prox

T2w-Lim PsTOPs

PrTOP,

TOPS denotes the construct of weakly Hausdorff limit spaces and continuous maps, where a limit space (X, q) is called weakly Hausdorff (or T2w) provided that the existence of a filter f on X converging to x, y e X implies {Q & FIL(X) (G, x) e q} = {H & FIL(X) | (U, y) € q}. "r" and "c" denote the corresponding bireflections and bicoreflections

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respectively. Definition 4.1. Let X be a set and Bx a B-set on X. A function H : Bx —> P(FIL(X x X ) ) is called a filterisotopy (or a filter isotopic operator) on Bx , and the pair (Bx ,n) a filterisotopic space if the following axioms are satisfied: (fstl) (fet2) (fst3) (fst4)

B 6 Bx and U <= /x(-B) with U C V € FIL(X x X) imply V € n(B); M(0) = 0; BiCB2€ Bx imply ^(5:) C ^(J52); x € X implies x x x € /x({x}).

i is defined by setting x := {T C X \ x € T}. U is called a B-uniform filter iff U e n(B). Now, let (BX,(J.) and (BY,rj) be filterisotopic spaces. A bounded map / is called a fit-map iff (fit) B € Sx and U e /j(J3) imply (/ x /)(W) 6 7j(/[B]), where (/ x f}(U] := {R C Y x Y \ (f x fi-^R] € U} with (/ x f)~l(R\ := {(x.x'JeXxXKAiJ.A^efl}. The fit-maps will also be referred to as the bounded maps preserving uniform filters. We denote by F-ISOTOP the corresponding category. Example 4.1. (1) For an s-neigbourhood space or a supertopological space (BX,Q) respectively, we define a filterisotopy on Bx by setting for each B&BX: He(B) := {U € FIL(X x X) \ Q(B) x Q(B) C U}, where 0(B)x 6(5) := {fl C X x X | 3Fi,F2 e 0(J3).fl 2 Fj xF2}. In the opposite direction let a filterisotopic space ( B x , r j ) be given. Define an underlying s-neighbourhood structure 6,, by setting: := {V C X | VW e rj(B). V x V e W} for each B € Bx . This leads us to a one-to-one correspondence between all sneighbourhood structures (supertopologies) on a B-set Bx and the so defined generated filterisotopies on it. Consequently, we obtain a full embedding of the category STOP into F-ISOTOP.

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LESEBERG (2) Consider a B-set Bx :- {0}U{{o:} | x e X], where X is a set. For a generalized convergence relation q in X [21] we obtain a filterisotopy on Bx by putting:

) := {U € FIL(X x X) | 3.F e FILpC). (.F.s) € q and jFx JFCW}. In the opposite direction each filterisotopy 77 on a B-set Bx has an underlying convergence relation p^ by setting: (J7, x) € p,, iff there exists IA € f?({x}) such that U C .F x .F. This leads us to a one-to-one correspondence between all generalized converge relations on Bx and the so defined discrete filterisotopies on it. Consequently one obtains a full embedding of the category GCONV into P-ISOTOP. Remark 4.1. It should also be noted that the concrete category F-ISOTOP is topological that means it satisfies the following conditions: (tCl) Existence of initial structures. For any B-set Bx , any family (BXi,Hi)i£i of filterisotopic spaces, and any family (/, : Bx —> BXi)i£i of bounded maps there exists a unique filterisotopy My-i on Bx which is initial with respect to the given data (Bx , f i , (BXi,p,i), /), i.e., such that for any filterisotopic space (S y ,/Lt) a bounded map

is a fit-map if for every i £ / the composite map fi° g '• (BY , JJL) —> (BXi,Hi) is a fit-map. (tC2) Fibre- Smalness. For any B-set Bx , the F-ISOTOP-fibre, i.e., the class of all filterisotopies on Bx , is a set. (tC3) Terminal Separator Property. For any set X with cardinality one there exists precisely one filterisotopy on Bx = {0, {x}}. Moreover, F-ISOTOP is extensional which means there exist universal one-point extensions, i.e., every F-ISOTOP-object A can be embedded via the addition of a single point oo into an object A* :— A U {00} such that the following holds: For every F-ISOTOP-morphism / : U —> A from a subspace U of an F-ISOTOP-object B into A the unique function /* : B —> A*

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defined by

( f(b) if b e [/; /*(&) == < [ oo otherwise is an F-ISOTOP-morphism. For basic literature concerning the above mentioned definitions the reader is referred to the book of G. Preufi [21]. Motivated by Example 4.1(2) in the following we only consider functions from a set X into the set P(FIL(X x X)) and define consequently: Definition 4.2. For a set X, a function fj, : X —> P(FIL(X x X)) is called an equiconvergence (equiconvergence operator) on X, and the pair (X, [A) is called an equiconvergence space iff (EC1) x 6 X, U £ n(x) and U C V € FIL(X x X) imply V € /z(x); (EC2) a; € X implies i x x £ p,(x). Elements U of /x(a;) are called uniform filters /x-equiconvergent to x, and x is called an equiform limit of U with respect to fj,. Given a pair of equiconvergence spaces (X,(j,), (Y,r)), a map / : X —> Y is called equicontinuous or, shortly, an ec-map iff (ec) x G X and U € n(x) imply (/x/)(W) € J?(/(i)); where (/x/)(W) :=

{^ c y x r (/ x /)-![-R] e W} with (/ x /)-![#] := {(x,x') e X x x |(/(a:), /(i')) efl}. The ec-maps will also be referred to as the maps preserving equiconvergent uniform niters. We denote by EConv the corresponding category. Example 4.2.

(1) With respect to Example 4.1(2) let (X,q) be a generalized convergence space. We obtain an equiconvergence operator on X by setting: pq(x) := {U € ¥\L(X xX)\3f&

FIL(X). (?,x) € q and

(2) For a preuniform convergence space (X, Jx) we obtain an equiconvergence operator on X by setting: (J,j(x) := Jx for each x £ X.

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Remark 4.2. We note that EConv can be considered as a full subcategory of F-ISOTOP. Theorem 4.1. The category GConv of generalized convergence spaces (and contiinuous) maps is isomorphic to a full subcategory which is bicoreflective in EConv. Proof. With respect to Example 4.2(1) in connection with Example 4.1(2) we obtain the desired isomorphism. Moreover we note that the induced equiconvergence operator nq satisfies additionally the definition of simple equiconvergence, which generally means (sec) U G r](x) implies there exists F G FIL(X) such that (J7, x) G pn and T x T C U. S-EConv denotes the full subcategory of EConv whose objects are the simple equiconvergence spaces. To show that S-EConv is bicoreflective in EConv, let (X, //) be an arbitrary equiconvergence space. We define a simple equiconvergence operator on X by setting: p,s (x) := {V G FIL(X x X) \ 3F G FIL(X). ( f , x) G gM and F x F C V} for each x G X. Then ps is an equiconvergence operator which is simple. The identity \\ : (X,/j,s) —> (X, /u) is equicontinuous, i.e., V G ns(x) implies there exists F G FIL(X) with (F, x) & q^ and ^x F C V. Moreover, we can find U e p(x) such that U C f x f, hence U C V follows which shows V is /u-equiconvergent to x. To finish the proof let (Y,r/) be a simple equiconvergence space and let / : (Y, r/) —> (X, n) be an equicontinuous map, we have to show that / : (Y, TI) —> (X, f i s ) is equicontinuous. V G r/(y) implies that we can find T e FIL(y) with (J7, y) € q^ and J7 x F C V, since 77 is simple by assumption. Moreover, there exists U G r/(y) such that W C J7 x J7, hence (/ x /)(W) G p>(f(y)} follows because / is equicontinuous. We have (1) /(J7) x /(J7) C /(V) such that (2) /(W) C /(J-) x f ( F ) . But this shows (/ x /)(V) G (is(f(x))

which has to be proven.

D

Theorem 4.2. The category PUConv of preuniform convergence spaces (and uniformly continuous maps) is isomorphic to a full subcategory which

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203

is bireflective in EG/onv. Proof. With respect to Example 4.2(2) we note that fij is a constant function which means that for each pair (x, x'} € X x X we have 77(0;) = TJ(X'). (Note that H j x ( x ) = J\ for each x € X.) Conversely let there be given such an equiconvergence operator 77. We set jCrj := (J{T?(x) x & X}. Then we obtain the desired isomorphism. C-EConv denotes the full subcategory of EConv whose objects are the constant equiconvergence spaces. To show that C-EConv is bireflective in EConv, let (X,fj.) be an arbitrary equiconvergence space. We define a constant equiconvergence operator on X by setting: /V) := {V € FIL(X x X) | 3x 6 X. V e n(x)} for each x' £ X. Then nc is a constant equiconvergence operator on X, and the identity mapping \x '• (X,fj.) —> (X,p,c) is equicontinuous. The universal property then can be easily verified. D Now, we will examine the concrete category EConv. The next theorem is along this line. Theorem 4.3. EConv is a topological category. Proof. For a class-indexed family (Xj,^j) i 6 / of equiconvergence spaces such that, for each i e /, fa : X —> Xi is a map from a set X to Xi, we define for each x £ X: H f - ^ x ) := (U 6 FIL(X x X) \ (/« x / 4 )(W) e W (/i(ar)) Vt e /}. Then [i.^r\ is the initial equiconvergence operator on X with respect to the given data. All other conditions for being a topological category are obviously satisfied. D Next, we will show that EConv has natural function space structures. Theorem 4.4. EConv is cartesian closed. Proof. For any pair (X,nx), (Y,/zy) of equiconvergence spaces let Yx denote the set Yx := {/ / : ( X , p , x ) —> (Y, py) is equicontinuous}.

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We define an equiconvergence operator /iy

by setting for each / 6 Yx :

HY* (/) := {U" € FIL(FX x Yx) \ Vx e X VU 6 nx(x). e(U x ZT) e where e(W x W*) denotes the filter generated by {e[U x U*} \ U £ U, [7* e W*} with e[t/ x t/*] := {e((z, *'),(/,/')) I (*,*') e t/, (/,/') e W*} = {(/(a?), /(^O) I (z,O e U , ( f , f ) e 17*}. This equiconvergence operator /iy is called uniformly continuous. The remaining verifications are left to the reader. D In the next we will show that EConv has universal one-point extensions; in other words,we will show that partial morphisms in EConv are representable. Theorem 4.5. EConv is extensional. Proof. Let (X, ^) be an equiconvergence space. Put X* := X U {00} with oo i X. For each U C X* x X* define [7* := U U (X* x {oo}) U ({00} x X*).

For each U 6 FIL(X x X) consider the filter U* := {U* U 6 U} on X* x X*. An equiconvergence operator /j," on X* can be defined by setting V € ^*(a;) iff x = oo or do x ob = V or there is some W 6 ^(x) with W* C V or {(oo, oo)}* € V, i.e., all filters on X* x X* ^"-converge to oo, and ob x ob //"-converges to all elements of X* while in any other case the convergence behaviour of V G FIL(X* x X*) is determined by its trace on X x X. D Corollary 4.1. EConv is a topological universe. The following diagram gives us a complete oversight of the just handled structures or spaces respectively. By TOPGEN we denote the category of topogenous spaces and continuous maps in the sense of Czazar [7] . QPROX is related to the quasiproximities of Fletcher and Lindgren,

205

SUPERTOPOLOGIES which can be found in Ref. [18].

SN

F-ISOTOP

NEAR

CONT

LOPROX

KConv

PROX

UNIF

CHY

KConvg

PROX

LIMg

TOPS

Last Sentence If we ornit Axiom (SN6) in the definition of a supernear space, then the so remaining semi-supernear spaces are a generalization of the sneighbourhood spaces, too. References [1] J. Adamek, J. Reiterman, Cartesian closed hull of the category of uniform spaces, Topology Appl. 19 (1985), 261-276. [2] B. Banaschewski, Extensions of topological spaces, Canadian Math. Bull., 7 (1964), 1-23. [3] A. Behling, Einbettung Uniformer Rdume in topologische Universen, Free University of Berlin (1992). [4] R. Beattie, H.-P. Butzmann, Convergence structures and applications to functional analysis, Dordrecht, London, Kluwer Academic (2002). [5] H. L. Bently, Nearness spaces and extension of topological spaces, Studies in Topology, Academic Press, NY (1975), 47-66. [6] H. L. Bently, H. Herrlich and E. Lowen-Colebunders, Convergence, J. Pure and Appl. Algebra 68 (1990), 27-45.

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[7] A. Czaszar, Foundations of General Topology, Pergamon Press (1963), Oxford-London-New York-Paris. [8] D. Doitchinov, A unified theory of topological, proximal and uniform spaces, Doklady Akad. Nauk SSSR 156 (1964), 21-24 (Russian); English transl.: Soviet Mathematics Doklady 5 (1964), 595-598. [9] D. Doitchinov, Compactly determined extensions of topological spaces, SERDICA Bulgaricae Mathematicae Publicationes 11 (1985), 269-286. [10] E. Hayashi, On some properties of a proximity, J. Math. Soc. Japan 16 (1964), 375-378; MR 31 #2708. [11] H. Herrlich, A concept of nearness, Gen. Top. Appl. 5 (1974), 191-212. [12] H. Herrlich, Cartesian closed topological categories, Math. Coll. Univ. Cape TownQ (1974), 1-16. [13] M. Husek, Categorical connections between generalized proximity spaces and compactifications, Contributions to Extension Theory of Topological Structures (Proc. Symp. Berlin 1967), Berlin (1969), 127-132. [14] V. M. Ivanova, A. A. Ivanov, Contiguity spaces and bicompact extensions, Dokl. Akad. Nauk SSSR 127 (1959), 20-22. [15] M. Katetov, On continuity structures and spaces of mappings, Comment. Math. Univ. Carolinae 6 (1965), 275-278. [16] M. W. Lodato, On topological induced generalized proximity relations I, Proc. Amer. Math. Soc. 15 (1964), no. 3, 417-422. [17] M. W. Lodato, On topological induced generalized proximity relations II, Pacific J. Math, no. 1, 131-135. [18] S. A. Naimpally, B. D. Warrack, Proximity Spaces, Cambridge (1970). [19] G. Preufi, Cauchy spaces and generalizations, Math. Japonica 38 (1993), 803-812. [20] G. Preufi, Semiuniform convergence spaces, Preie Universitat Berlin, Fachbereich Mathematik, Serie A, Preprint No. A-24-94, 1-32. [21] G. Preufi, Foundations of Topology. An Approach to Convenient Topology, Kluwer (2002). [22] G. Preufi, Topologically generated and uniformly generated spaces, Freie Universitat Berlin, Fachbereich Mathematik, Serie A, Preprint No. A-0602, (2002). [23] J. Slapal, Net spaces in categorical topology, Ann. New York Acad. Sci. 803 (1996), 393-412. [24] Y. M. Smirnow, On the completeness of proximity spaces, Dokl. Akad. Nauk SSSR 88, 761-794 (in Russian); MR 15 #144. [25] A. D. Taimanov, On extensions of continuous mappings of topological spaces, Mat. Sbornik N. S. 31 (1952), 459-463. [26] W. L. Terwilliger, On Contiguity Spaces, Thesis, Washington State University (1965). [27] A. Tozzi, O. Wyler, On categories of supertopological spaces, Acta Universitatis Carolinae-Mathematica et Physica 28(2) (1987), 137-149. [28] O. Wyler, On convergence of filters and ultrafilters to subsets, Lecture Notes in Comp. Sci. 393 (1989), 340-350.

INTRODUCING LAGOIS CORRESPONDENCES

AUSTIN MELTON Departments of Computer Science and Mathematical Sciences Kent State University Kent, Ohio 44242, USA E-mail: [email protected] Usually, when Galois connections are generalized categorically, they are generalized to adjoint situations. Unfortunately, in this generalization many interesting properties are lost. Recently, Galois connections have been generalized in a concrete category setting to form Galois correspondences. Fortunately, this generalization preserves many Galois connection properties. Also, recently, a counterpart to Galois connections has been defined and studied. This counterpart is Lagois connections, and in this paper, we generalize Lagois connections to Lagois correspondences. Mathematics Subject Classifications (2000): Primary: 06A15 Keywords: Galois connection, Lagois connection, Galois correspondence, Lagois correspondence

1. Introduction Galois connections - also called residuated/residual maps [3] - have been studied and used extensively in both mathematics and computer science. See [5] and [12] for many examples. In the early 1990's, a counterpart to Galois connections, called Lagois connections [11, 12], was discovered. In a Galois connection, the composite maps behave differently with respect to the order relations in the partially ordered sets. One composite is an increasing map and the other is a decreasing map. In a Lagois connection either both composites are increasing maps or both are decreasing. This sameness of the composite maps with respect to the order relations seems to allow Lagois connections to be better models than Galois connections in many computer science applications. See [12]. Historically, when Galois connections with order-preserving maps were generalized to a categorical setting, they were generalized to adjoint situations. This generalization is natural because when the partially ordered 207

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sets, upon which a Galois connection is defined, are considered as categories the order-preserving maps become an adjoint pair. Unfortunately, not many Galois connection properties are preserved in this generalization. In [8] , Herrlich and Husek show that when Galois connections are generalized to Galois correspondences in a concrete category setting, most of the interesting Galois connections properties are preserved. In this paper, we generalize Lagois connections to Lagois correspondences. Much of the inspiration for this current paper comes from [8] . It should be pointed out that in [8] Herrlich and Husek actually define two generalizations of Galois connections. In addition to Galois correspondences, they also define Galois adjunctions. Further, they use the expressions "Galois connections of the third kind" and "Galois connections of the fourth kind" for "Galois correspondences" and "Galois adjunctions" , respectively. Also, in [8], polarities, which are special Galois connections, are also called Galois connections of the first kind, and Galois connections, themselves, are called Galois connections of the second kind. We begin by reviewing Galois connections and Lagois connections and by discussing their similarities and differences. Then we review Galois correspondences before defining and investigating Lagois correspondences. 2. Galois Connections and Lagois Connections As the name "Galois connection" suggests, Galois connections are related to the work of E. Galois. The relationship which Galois discovered between field extensions and groups of automorphisms is an example of a Galois connection. This example is given in 2.2. In 1944, O. Ore [13] defined a Galois connection, which he called a "Galois connexion" , as follows. Definition 2.1. Let (P, <) and (Q, <) be partially ordered sets, and let / : P —> Q and g : Q —> P be maps. (/, (P, <), (Q, <),#) or simply (/,#) is a Galois connection if / and g are order-reversing maps and idp < g f a and Example 2.2. Let E : F be a field extension, and let P be the set of fields L such that "I.e., for each p 6 P,p < gf(p) because idp(p) = p < gf(p). b There is an alternative way to define Galois connections. One can replace the conditions idp < gf and idq < fg with the condition that Vp e P and Vg € Q, p < g(q) iff <7 < /(p).

LAGOIS CORRESPONDENCES

209

E is a subfield of L and L is a subfield of F. Order ~P by subset inclusion. Let G be the group of field automorphisms of F that fix E pointwise, and let Q be the set of subgroups of G also ordered by subset inclusion. Define / : P - Q by

and define g : Q —> P by

g(H) = {z€ F\Vh 6 H, h(z) = z}. (/, g) is the Galois connection that arises from Galois' work in Galois theory. The maps in Galois' example and the maps in Ore's definition are orderreversing maps. In 1953, J. Schmidt [15] introduced order-preserving Galois connections. Order-preserving Galois connections can be obtained by beginning with an Ore-defined Galois connection and reversing the order on Q, i.e., by using the dual order on Q. Definition 2.3.

Let (P, <) and (Q, <) be partially ordered sets, and let / : P —> Q and g : Q —> P be order-preserving maps. (/, (P, <), (Q, <),#) or simply (/,#) is an order-preserving Galois connection if idp < gf and fg < idq.c An interesting difference between order-reversing and order-preserving Galois connections is in the case of order-reversing maps the images g[Q] and /[P] are anti-isomorphic whereas with order-preserving maps the images g[Q] and /[P] are isomorphic partially ordered sets. An advantage of order-preserving Galois connections, when compared to order-reversing ones, is order-preserving Galois connections can be composed and the result is another order-preserving Galois connection. Order-preserving Galois connections are studied in lattice theory under the name residuated/residual maps. In Definition 2.3, / is the residuated map, and g is the residual one. The study of residuated and residual maps is residuation theory, and the standard reference is Residuation Theory by T.S. Blyth and M.F. Janowitz [3]. Prom this point on in this paper, all Galois connections will have orderpreserving maps. c

There is also an alternative definition for order-preserving Galois connections. The conditions that gf is an increasing map and fg is a decreasing map are replaced by the condition that Vp e P and Vg £ Q, p < g(q) iff f(p) < q.

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If one looks at the computer science Galois connections in [10] — in particular, at the compiler and data coercion examples — one notices a common characteristic. In each example, one of the maps is injective. These are not examples of general Galois connections. While writing [10] and for a time thereafter, we (the authors of that paper - Melton, Schmidt, and Strecker) tried to find more general examples of Galois connections in computer science; in particular, we were trying to generalize the examples in that paper to Galois connections in which neither map would be injective. We did not succeed. The generalized examples are found in [12]. However, the generalized examples are not Galois connections; they are Lagois connections. It was in trying to develop a mathematical construction that would model an example like the compiler example in [12] that Lagois connections were discovered. Definition 2.4. Let (P, <) and (Q, <) be partially ordered sets, and / : P —> Q and g : Q —> P be order-preserving maps. Then (/, (P, <), (Q, <), Q and g : Q —> P be order-preserving maps. (/, (P, <), (Q, <),
LAGOIS CORRESPONDENCES

211

(/> (Pi <)' (Qi <)>#) is either a Galois connection or a Lagois connection. Though the properties are very similar, the proofs for a Galois property and its equivalent Lagois property are often different. The Galois proofs are found in many places, including [3, 10, 13, 14]; Lagois proofs are in [12]. Prop 2.1. Let (/, (P, <), (Q, <),g) be a Galois connection or a Lagois connection. (1) / and g uniquely determine each other. (2) p € P is a fixed point of gf iff p 6 g[Q], and q e Q is a fixed point (3) / is injective iff g is surjective iff gf = idp and g is injective iff / is surjective iff f g = idq. (4) g[Q] and f[P] are isomorphic partially ordered sets, with the restrictions

being order isomorphisms, and these two restricted maps are inverses to each other. (5) If P and Q are complete lattices, then so are g[Q] and f[P] though they need not be sublattices. Here is an additional property for Galois connections. Prop 2.2. Let (f,(P,<),(Q,<),g) inverses of each other.

be a Galois connection. / and g are quasi-

The quasi-inverse conditions are included in the definition of Lagois connections so that Lagois connections properties would be similar to Galois connection properties. The next example shows that having the quasiinverse conditions or something equivalent is needed. Example 2.8. Let P - {a, b} with a < b, and let Q = {a1, b'} with a' < b'. Define / : P -> Q by /(a) = a' and /(&) = b', and define g : Q -> P by g(a') = g(b') =

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b. f and g are order-preserving, and idp < gf and idq < f g . However, g[Q] and f[P] are not isomorphic partially ordered sets and fgf ^= f . Prop 2.3. Let (P, <) and (Q, <) be partially ordered sets, and let / : P —> Q and g : Q —> P be order-preserving maps with idp < gf and idq < f g . Then the following conditions are equivalent: / and g are quasi-inverses of each other

and p £ P is a fixed point of gf iff p e g[Q] and q e Q is a fixed point of fgiSq€ f[P}This last proposition means that we could have defined Lagois connections by replacing the condition that / and g are quasi-inverses of each other with the conditions that p € P is a fixed point iff p € g[Q] and q 6 Q is a fixed point iff q € f[P}3. Galois Correspondences and Lagois Correspondences Most of the background material and results on concrete categories and Galois correspondences in this section are from [1] and [8]. The underlying idea for the proofs in Theorem 3.13 seems to be new.d Definition 3.1. Let A and B be categories, and let H : A —> B be a functor. H is said to be faithful if whenever h, k : X —> Y 6 Mor(A) and H(h) = H(k), then h = k. Definition 3.2. Let X be a category. A concrete category over X is a pair (A, U) where A is a category and U is a faithful functor. U is said to be an underlying or forgetful functor. d

The reader who compares this paper with [1] and [8] will find some differences in the order of presentation of / and g and of F and G. The presentation in this paper is consistent with the traditional order for Galois connections. In the [1] and [8], the order is switched for purposes of categorical consistency. Since the main purpose of this paper is to generalize Galois and Lagois connections, the traditional order for Galois connections is used.

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Definition 3.3. Let (A, U) and (B, V) be concrete categories over X, and let F : A —> B be a functor. F is said to be a concrete functor if it commutes with the forgetful functors U and V, i.e., if U = VF. Prop 3.1. A concrete functor is determined by its values on objects in that if F and F' : A -> B are concrete functors and if F(A) = F'(A) for each A <5 Ob(A) then F = F'. Given a concrete category (A, U) over X, the forgetful functor U partitions O6(A) into equivalence class fibres where for A, A' 6 O6(A), A and A' are in the same equivalence class, i.e., in the same fibre of U, iff U(A) = U(A'). We can define a preorder on Ob(A) by defining the preorder on each fibre.6 Prop 3.2. Let (A, U) be a concrete category over X, and let A, B 6 <96(A). Define A B € Mor(A) with Uh = iduA< is a preorder on the class O6(A). Prop 3.3. Let F : (A, U) —» (B, V) be a concrete functor over X. F is orderpreserving. Definition 3.4. Let (A, U) and (B, V) be concrete categories over X, and let F : A —> B and G : B —> A be concrete functors. (F, (A, [/), (B,V),G) or simply (F, G) is said to be a Galois correspondence over X if idob(A) < GF and FG < Definition 3.5. Let (A, U) and (B, V) be concrete categories over X, and let F : A —> B and G : B —> A be concrete functors. (F, (A, U), (B,V),G) or simply (F, G) is said to be a Lagois correspondence over X if < GF and idOb(B] < and FGF = F and GFG = G. e

lt should be noted that in this section, we generalize pre- and partial orders from sets to classes.

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MELTON

If (F, (A, U), (B,V),G) is a Galois or Lagois correspondence, then F and G respect the fibres of U and V as described in the following lemmas. Lemma 3.6. Let (F, (A, U),(B,V},G) be a Galois or a Lagois correspondence. If A, A' e Ob(A.) and if A and A' are in the same fibre of U, then F(A) and F(A') are in the same fibre of V. Similarly, if B,B' £ Ob(B) are in the same fibre ofV, then G(B] and G(B') are in the same fibre o f U . Lemma 3.7. Let (F,(A,U),(B,V),G) be a Galois or a Lagois correspondence. If A, A' € Ob(A) and if A and A' are in different fibres of U, then F(A) and F(A') are in different fibres ofV. Similarly, if B,B' e Ob(B) are in different fibres of V, then G(B) and G(B') are in different fibres of U. Example 3.8. Let (/, (P, <), (Q, <)> X by U(p) = X for each p G O6(P), and similarly, define V : Q —> X by V(q) — X for each q G O6(Q)- If (/>#) is a Galois connection, then (F, (P, <), (Q, <),G) is a Galois correspondence and if (/, g) is a Lagois connection, then (F, (P, <), (Q, <),C?) is a Lagois correspondence, where F and G are / and g, respectively, extended to morphisms. The Galois version of 3.8 is in [1]. Example 3.9. Let (F, (A, U), (B, V), G) be a Galois or a Lagois correspondence over X. Let X e X such that U~l(X) and V~1(X) are nonempty. Let P = U~1(X), and let Q = V~l(X}. Define < on P and Q by restricting the orders on Ob(A) and O6(B), respectively. Then if (F,G) is a Galois or Lagois correspondence, (F\p, (P, <), (Q, <),G\Q) is a Galois or Lagois connection, respectively. Considering these examples, and noting that the functors of the Galois and Lagois correspondences respect the Galois and Lagois connections defined on these fibres, many Galois and Lagois connection properties immediately generalize to "object properties" of Galois and Lagois correspondences. Further, since concrete functors are determined by their actions on

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objects, many Galois and Lagois connections properties immediately generalize to Galois and Lagois correspondence properties. These observations will be the basis of our proofs in Theorem 3.13. We need an additional property to help us generalize Galois and Lagois connection properties to Galois and Lagois correspondences. Currently, the orders on the object classes of the concrete categories are only preorders. The additional property, called amnesity, will give us partial orders. Definition 3.10. Let A and X be categories, and let U : A —> X be a functor. U is said to be amnestic if for every A-isomorphism /, whenever Uf is an X-identity, / is an A-identity. Alternatively, we can make the following definition. Definition 3.11. A concrete category (A, U) is said to be amnestic if the preorder defined by X < Y iff UX = UY and there exists h : X —> Y € MorA with Uh = idux is a partial order. Prop 3.4. The two definitions of amnesity are equivalent. Notation 3.12. Let (F, (A, U), (B, V), G) be a Galois or Lagois correspondence. Let A* be the full subcategory of A such that the objects of A* are the images under G of objects in B, i.e., O6(A*) = (G(B) : B & Ob(B)}, and let B* be the full subcategory of B such that Ob(B*) = {F(A) : A e Ob(A)}. Further, let U* be the restriction of U to A*, and let V* be the restriction of V to B*. Also, let F* and G* be the restrictions of F and G to A* and B*, i.e., F* : (A*,[/*) -> (B*,V*) and G* : (B*,V*) -> (A*,U*). Theorem 3.13. Let (F,(A,U),(B,V),G) be a Galois correspondence or a Lagois correspondence over X with (A, U) and (B, V) being amnestic concrete categories. (1) F and G uniquely determine each other. (2) A E O6(A*) iff A = GF(A) and B & Ob(B") iff B = FG(B)

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(3) F is injective iff G is surjective iff GF = id A , and G is injective iff F is surjective iff FG = id-s • (4) F* : (A*, 17*) -> (B*, V*) and G* : (B*, V*) -> (A*, [7*) are concrete isomorphisms that are inverse to each other. Proof: (1) Suppose that (F,G) and (F,G') are Lagois correspondences. Let B € O6(B). Consider the Lagois connections (F\v--nV(B)),G\v-i(V(B))) and (F\u-i(V(B)),G'\v-i(v(B)))Since the maps of a Lagois connection determine each other, then G(B) = G|v-i (V(B ))(S) = G'\v-1(vm(B) = G'(B). Therefore, G = G' on Ob(B), and hence, G = G'. The other results can be proven by using the corresponding results for the underlying Galois and Lagois connections. Theorem 3.14. (cf. [1],[8]) Let (F, (A, [/), (B, V), G) be an amnestic Galois correspondence over X. (1) FGF = F and GFG = G and (2) A* is reflective in A and B* is coreflective in B. Theorem 3.15. Let (F, (A, [/), (B, V), G) be an amnestic Lagois correspondence ewer X. Then A* is reflective in A and B* is also reflective in B. Proof: Let A e Ob(A). GF(A) e Ob(A*), and since A < GF(A), there exists r : A -> GF(A) e Mor(A) with U(r) = idu(A). Let A* £ A*, and let / : A -> A* e Mor(A). Since A* € A*, then 4* = GF(A*), and thus, GF(/) : GF(A) -> A*. /,GF(/) o r : A -> A*. [/(/) = VF(/) = VFGF(f) = UGF(f) = UGF(f) o idu(A} = UGF(f) o U(r) = U(GF(f) o r). Since U is faithful, / = GF(/)or. Suppose /i : GF(A) -> A* £ Mor(A) with hor = f. Then C/(/i o r) = U ( G F ( f ) o r ) , and U(h)oU(r) = U ( G F ( f ) ) o U ( r ) . Hence, U(h)oidu(A} = U(GF(f)) o idy^), and 17(/i) = U(GF(f)). It follows that /i = GF(/). Therefore, A* is reflective in A.

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References [I] J. Adamek, H. Herrlich and G.E. Strecker. Abstract and Concrete Categories. John Wiley & Sons, New York, 1990. [2] G. Birkhoff. Lattice Theory, 1st ed., Amer. Math. Soc., Colloquium publications, v. 25, New York, 1940. (3rd ed, 1967, Providence, RI) [3] T. S. Blyth and M. F. Janowitz. Residuation Theory. 1st ed., Pergamon Press, Oxford, 1972. [4] H. Crapo. Ordered sets: retracts and connections. J. of Pure and Applied Algebra, 23 (1982) pp. 13-28. [5] M. Erne, J. Koslowski, A. Melton, G.E. Strecker. A Primer on Galois Connections, Papers on General Topology and Applications, Annals of the New York Academy of Sciences, 704 (1993), 10 3-125. [6] G. Gierz et al. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, 1980. [7] H. Herrlich and M. Husek. Galois connections. Lecture Notes in Computer Science, No. 239, Springer-Verlag, Berlin, 1986, pp. 122-134. [8] H. Herrlich and M. Husek. Galois connections categorically. J. of Pure and Applied Algebra, 68 (1990) pp. 165-180. [9] C.A.R. Hoare. The Mathematics of Programming. Clarendon Press, Oxford, 1986. [10] A. Melton, D.A. Schmidt, and G.E. Strecker. Galois connections and computer science applications. Lecture Notes in Computer Science, No. 240, Springer-Verlag, Berlin, 1986, pp. 299-312. [II] A. Melton, B.S.W. Schroder, G.E. Strecker. Connections. Lecture Notes in Computer Science, No. 598, Springer-Verlag, Berlin, 1992, pp. 492-506. [12] A. Melton, B.S.W. Schroder, G.E. Strecker. Lagois connections - a counterpart to Galois connections, Theoretical Computer Science, 136 (1994), 79-107. [13] O. Ore. Galois connexions. Transactions of the American Mathematical Society, 55 (1944), pp. 493-513. [14] G. Pickert. Bemerkungen iiber Galois-Verbindungen. Archiv der Mathematik, 3 (1952), pp. 285-289. [15] J. Schmidt. Beitrage zur Filtertheorie, H. Mathematische Nachrichten, 10 (1953), pp. 197-232. [16] P. Wegner. Programming language semantics. Courant Computer Science Symposium 2, Prentice-Hall, Englewood Cliffs, New Jersey, 1972, pp. 149248.

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ON (B, B)-PROJECTIVITY

HELGA OLTMANNS Willerstr. 7 26123 Oldenburg Germany E-mail: Helga. Oltmanns Qswb-Enordia. de ULRICH KNAUER Institut fur Mathematik Carl von Ossietzky Universitat Oldenburg 26111 Oldenburg Germany E-mail: knauer@uni-oldenburg. de VALDIS LAAN AND MATI KILP* Institut fur Mathematik Universitat Tartu EE2400 Tartu Estonia E-mail: [email protected] E-mail: [email protected]

We consider concrete categories C with free objects, surjective epimorphisms and coproducts and give two as far as we know new characterizations of projective objects. For this we use a concept of projectivity which restricts the lifting property to certain families of epimorphisms and even to one special epimorphism. The respective equivalences are formulated in categories which fulfill a condition (*). Mathematics Subject Classifications (2000): 18G05, 20M30, 13C10, 16D40 Keywords: projectivity, category, act, module, semimodule.

'Research supported by Estonian Science Foundation grant no. 4557 and Deutsche Forschungsgemeinschaft (DFG). 219

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An object A of a category C is called projective if in the following situation in C every homomorphism / can be lifted to B relative to any epimorphism 9-

A f /

i. e. there exists tp such that / = gtp. Consider in C morphisms TT : A —> B and 7 : J5 —> A with tr"/ — ids, which means in particular that TT is an epimorphism and 7 is a monomorphism. Then we call TT a retraction, B a retract of A, and 7 a coretraction, cf. for example [2] or [3]. It is well known that if C is a concrete category with free objects and surjective epimorphisms, then an object A of C is projective if and only if it is a retract of a free object. In the following we will prove two further conditions equivalent to projectivity, first for the category of right S-acts, where 5 is a monoid, and then for categories which fulfill some condition (*). In [5] an object A is called (B, C) -projective in C, if A is projective relative to all objects B and C and all epimorphisms g : B —» C. So if there are no restrictions on B and C then projective and (B, C)-projective are the same. Restrictions will lead to various forms of weak projectivity, cf. [4] and [5] for the category of right 5-acts. These projectivities are dual to weak injectivities, cf. [3]. For other needed concepts, not defined here, compare [2] or [3]. \. (B,,b£?)-projective right acts over monoids By S we denote a monoid with identity Is or just 1. A right S-act AS is a set A on which S acts unitarily from the right, i.e., a(st) = (as)t, and al — a for a e A, s, t e S, 1 £ S. If AS and BS are right 5- acts and / : AS —> BS is a mapping, then / is called a homomorphism if /(as) = f ( a ) s for a € A, s £ S, epimorphisms are surjective homomorphisms. By Act —S we denote the category of right 5-acts. The coproduct in this category is the disjoint union, denoted by ]J, with the injections, and free objects are the coproducts of copies of SsIn the following we shall consider in particular (B, B)-projectivity in Act -S.

(B,B)-PROJECTIVITY

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Theorem 1.1. Take AS e Act —5. The following are equivalent (i) AS is projective. (ii) AS is (B,B)-projective, i. e., for every epimorphism g : BS —* BS and homomorphism f : AS —> BS there exists a homomorphism : AS —> BS such that f = gip. (Hi) Any canonical injection of AS into the coproduct B = (LLeN^)II(LLeN F ( A ))> where F(A} is the free ridht S-act with the basis A, can be lifted with respect to every epimorphism g:B-*B. (iv) AS is the retract of a free act. Proof. We only have to prove that (iii) implies (iv). Let AS be a right S-act satisfying (iii) and let F(A)s be the free right S-act with the basis A. Then there exists an epimorphism a : F(A)s —> AS- Consider in Act — S the following diagram

n

) II (U n6N F»(A)) -i- Qln6N A») U (U

where the upper index represents a numbering of the copies of A and F(A). Here the injection i:A —> (U n€N ^") LI (ILeN Fn(A)) is given by i(a) = a1 e A1. The epimorphism

n

ii ^ u u r-w - n ** n n

\n6N

is given by n

-+An+\ g(an) = an+\ n > 1, 1 * A, > Fn~i(A), g(fn) = /"-1, n > 1.

Here /*, i e N, denotes the element / e F(,4)s in the i-th copy F^A) of F(A)S in the coproduct IIneN^ n (^) and «* denotes the element a e As in the i-th copy of ^4S in the coproduct IJneN^"- ^y ( in )> there exists a homomorphism
(]JneN A") U (LL € N F "(^)) with ff¥> = t. This implies ^?(A) C F1(A). ThusfflFJCA)1^= aip = id^, i-e. As is a retract of the free act F1(A). D

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Definition 1.1. We call an object basicly protective if it has the property of Condition (iii) in the previous theorem. 2. (B, B)-projectivity in categories Let / be an index set and Bi, i € /, objects of a category C. Let ((ui)i£i,B) = LI ie /Bi be the coproduct of the objects 5,, let a : I —> I be a bijection and let on : Bi —> j5CT(i) be isomorphisms for all i e /, with the possible exception of a CT -i(j 0 ), where i0 e / is a fixed index. Denote by g := {(u«r(i)at)ie/) : B —» B the coproduct induced morphism. Condition (*): If in this situation there exists a morphism y> : Bi0 —» B such that g(p = uio, then there exists t/j : Bio —> B^-i^ such that

Theorem 2.1. Suppose that C is a concrete category with free objects, surjective epimorphisms and coproducts, which satisfies Condition (*). For every object A of C the following are equivalent (i) A is protective. (ii) A is (B, B) -protective, i. e,, for every epimorphism g : B —* B and morphism f : A —» B there exists a morphism ip : A —> B such that f=9V(iii) A is basicly projective, i. e., the injection of A into the coproduct B = (UnsN ^) II (UneN F(A)) , where F(A) is a free object such that A is its epimorphic image, can be lifted with respect to every epimorphism g : B —> B. (iv) A is the retract of a free object. The proof, presented here, is a direct generalization of the proof in [5] cited in the previous section for the category of acts over monoids. Proof. Again we have to prove only that (iii) implies (iv). Let A satisfy (iii). For the object A there must exist a free object F(A) and an epimorphism TT : F(A) —» A. We show that A is a retract of F(A).

(B,B)-PROJECTIVITY

223

Take I := N X {1,2},

S(n,2) := for every n G N, and consider the coproduct

Define a bijection a : / —» / as follows: |> + l , f c ) , f c = l, a ( n , A ; ) : = < (1,1), (n, *;) = (!, 2), [ ( n - l , f c ) , fc = 2 , n > 2 . Let o.i : Bj —» SCT(i) be isomorphisms if i ^ (1,2), set a(i,2) := ^ and to :=(!,!)•

R

a(1

''J p

°

•0(1,1) -*• -0(2,1) -P *• -0(3,1)

o

°£L!1 R

°(3'2) p

'" -0(1,2)-*- 0(2,2) •*- -0(3,2) ' • •

T— "(1,2)

Now for the coproduct induced morphism g = ((f B we have 5Ui = u a(i )ai

(1)

for every i G /, so, in particular,

We show that g is an epimorphism. Suppose that pg = qg for some morphisms p, q : B —» X, Since TT is an epimorphism,

implies pw(i,i) = B such that "(1,1) = 9
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By Condition (*) where i0 = (1, 1), there exists V> : #(1,1) -* .B F(A), such that

Define morphisms u>j : Bi —> j4, i e /, by

,2) •••«(«,2), A = 2,ra > 2. Let /i := {(u>i)jg/) : B —> >1 be the coproduct induced morphism. Then hui — Wi for every i £ I. In particular,

Finally, the calculation

shows that A is a retract of F(A).

D

3. Applications Condition (*) is not very specific. We do not know any particular classes of categories which can be associated with Condition (*). At least the following categories satisfy Condition (*): (1) The category Act —S of right acts over a monoid S. Recall that the coproducts in this category are disjoint unions. For a, cti, IQ and g as in the assumption of Condition (*) we get gm = ua^ai for every i € 7. Hence

g(b) = (gui)(b) - (u^a^b) = a* (6) for every i E I and 6 e Bi, that is, g(Bi) C Ba^. If for some (p : Bi0 —> B we have g
BO—I^ with y> = u,,-i(io)i(j. (2) The category of right acts over a semigroup (without identity), using the argument corresponding to that in Act —S, compare also [6].

(B,B)-PROJECTIVITY

225

(3) The category of sets. In this category all objects are free, hence also projective and basicly projective. Condition (*) can be verified exactly as in the case of Act —S. (4) A complete upper semilattice considered as a category. Condition (*) follows from the fact that the only way to choose a's is to take them all equal to identity. (5) The category of right semimodules over a semiring R. Let Bi, i € /, be right semimodules over a semiring R, and CT, o^, IQ and g as in the assumption of Condition (*). Let P = Ydei^i ^e tne cartesian product of sets Bi, i G I, with componentwise operations, and TTj : P —> Bi the projections. This is the categorical product of semimodules (cf. [1], Example 14.28) and the mappings TTJ are semimodule homomorphisms. Then B = {a; e P | 7Tj(o;) ^ 0 forfinitely-manyindices i e /} is the coproduct together with the injections u, : Bi —* B which take elements bi of BI to sequences, where the only non-zero component is the i-th component 6j. Note that TTJUJ = id^ for every i & I. Consider the coproduct induced homorphism g = ((ua^ai)i^i), denote k0 := a~1(i0), then u(k0) = i0. If i € / \ {k0} and bt 6 Bi \ {0} then 0 ^ a;(6;) = Tr^u^a^&i) = n^^gu^bi) e B^), because a^ is bijective. Suppose that g(p — Ui0 for a homomorphism Bk0 by

then for b e Bia and
and hence Uk0i> = (p. To prove that
Then
and

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=g

All summands, where j / fc0, are sequences with precisely one nonzero component at different places. Hence the sum cannot be Ui0(b) if J contains elements different from fco. Consequently J = {fc0} and ip(b) = ukaKko(p(b) e uko(Bko). Hence the Condition (*) is fulfilled. According to ([1], Proposition 15.15) a homomorphism / : M —> N of left .R-semimodules is surjective if and only if it is an epimorphism and /(M) is a subtractive subset of N. So Theorem 2.1 does not apply directly in this category. However, the proof of this theorem still works, because every semimodule is an epimorphic image of a free semimodule and projective semimodules are exactly the retracts of free semimodules (see [1], Propositions 17.11 and 17.16). (6) The category Mod — R of right modules over a ring R with the argument analoguous to the category of semimodules since coproducts have the same structure. References [1] J. S. Golan, Semirings and their Applications, Kluwer Academic Publishers, Dordrecht 1999. [2] H. Herrlich, G. Strecker, Category Theory, Allyn and Bacon, Boston 1973. [3] M. Kilp, U. Knauer, A. V. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin 2000. [4] U. Knauer, H. Oltmanns, Weak projectivities for S-acts, in : K. Denecke, H.J. Vogel (Hrg.), General Algebra and Discrete Mathematics, 143-159, Shaker, Aachen 1999. [5] H. Oltmanns, Homological Classification of Monoids by Projectivities of Right Acts. Doctoral Dissertation, Oldenburg 2000. [6] S. Talwar, Morita equivalence for semigroups, J. Austral. Math. Society, Ser A 59 (1995), 81-111.

ON COALGEBRAS WHICH ARE ALGEBRAS

HANS-E. PORST AND CHRISTIAN DZIERZON* Department of Mathematics University of Bremen, 28334 Bremen, Germany E-mail: [email protected] E-mail: dzierzon@math. uni-bremen. de

The category CoalgS of coalgebras with respect to a (bounded) signature S is known to be locally finitely presentable (see [1]). We strenghten this result by showing that CoalgS even is a presheaf category. Moreover, we give a presentation of this category as the category of all algebras of some (many-sorted) signature (without any equations). Mathematics Subject Classifications (2000): 18B99, 08A60 Keywords: S-coalgebras, S-labelled trees, presheaf categories

E-coalgebras, i.e., coalgebras with respect to a polynomial endofunctor HX on Set, H-z(X) = ]J Sn x *"> with S = (^n)n
known to be intimately related to tree structures*. On the one hand the set T£ of all E-labelled trees (see 1 below) is the underlying set of a terminal object in CoalgE, the category of S-coalgebras; on the other hand, each Elabelled tree t is a E-coalgebra At in its own right (see Definition 3 below). The structural importance of the family of tree coalgebras A t , t £ TE, already emerged in [1] where this family was shown to be a strong generator of finitely presentables in CoalgE. We are going to show in this note that this family even is an absolute generator, i.e., that the horn-functors determined by its members even preserve all colimits, which then leads to a representation of CoalgE as a presheaf-category (see also [5], where completely different methods have *we are grateful to J. Adamek for fruitful discussions on the subject a The results of this note generalize immediately to the case of A-ary signatures, where A is an infinite regular cardinal. 227

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DZIERZON

been used to establish such a presentation). The particular structure of the full subcategory spanned by the tree coalgebras then even allows for a simple explicit description of this presheaf category as a category of unary algebras without equations. We start by briefly recalling some basic concepts. 1 A S-labelled tree is a partial function

t: w* ->E whose domain of definition, Deft, has the following two properties: (i) Deft contains the empty word e and is prefix-closed, i.e., if uv e Deft then u e Deft, and (ii) if i\ . . . ik 6 Deft and t(i\ . . . ik) has arity n, then for all j we have ii . . . ikj 6 Deft iff j < n. Let i be a E-labelled tree and w € Deft. By t(w—) we denote "the subtree of t with root w" , i.e., the tree with v e Deft(w-) <=>wv£ Deft and then t(w—)(v) = t(wv). If w = i 6 w n Deft, t(i— ) is a "maximal subtree" of t. The set TS of all S-labelled tree becomes a E-coalgebra TE by means of the action a-£ defined by = ((t(l-), t(2-),...,t(n-)), t(e)) where t(e) € Sn; i.e., the action essentially assigns to a tree its family of maximal subtrees. 2 Given a E-coalgebra C = (C, ac) one can define, for all c £ C, trees t c 6 T£ and elements cw G C (w 6 Deft c ) inductively as follows • ce : = c and t c (e) = a <£> ac(c) = ((ci)j< n , t c then is a homomorphism C —> TS and, in fact, the only one (see [3]). Thus Ts is terminal in CoalgS.

COALGEBRAS WHICH ARE ALGEBRAS

229

3 Given t e TE the tree coalgebra At is denned as follows: At = (Deft,a t ) with at(w) = ((wl, . . . , wri), a) with
with ft(e) = c, moreover here t and h are uniquely determined (t is the image of c under the unique homomorphism (C, ac) —»• (Ts, as), i.e., t = £ c ). Denote, for a coalgebra C = (C, a c ), the map sending c & C to h: At -^ C with ft(e) = c, by
From this one concludes, since U is faithful and conservative: 4 Proposition The family (At)teT s is strongly generating, i.e., the family of horn-functors hom(At,-), < e T s is jointly faithful and jointly reflects isomorphism. 5 Proposition hom(A t , — ) preserves colimits, for each t € Tsb. Proof Since the functor U : CoalgS —> Set is known to preserve colimits, the Corollary shows that, for each colimit cocone (A* : P, —> D) i€ / in CoalgS, the canonical map colim/]J t hom(A t) Di) —» U t hom(A t ,D) is bijective. If now colim/ horn (A t , Oj) -^ colim/ ]J hom(At, D») b

This is a special instance of the following more general result: Let, for some functor U: A —> B and some B-object B, the family of A-objects (Aj)/ be multifree on B w.r.t. U; if U preserves colimits and B is an absolute generator, then so is the family (Aj)/. (Use the equivalence ]J7 hom(Aj, -) ~ hom(B, -) o U.)

230

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denotes the canonical bijection resulting from commutation of colimits, it will be enough to prove that (with the canonical maps A*). Ao V = ]J(A*: colimhom(A t ,Di) -> hom(A t ,D)) t This can be read off the following commutative diagram (where Ht is short for horn (A t —))

where the maps indexed by t are coproduct injection, and the maps indexed by i form the respective colimit cones. D 6 Proposition hom(A t ,—) preserves kernel pairs, for each t e Ts. The family (hom(A t , —))tgr s collectively reflects kernel pairs. Proof Preservation is clear. Let now p, q: phisms such that, for each t e TS,

C be a pair of homomor-

hom(A t ,p), horn(A t , hom(A t ,C) is a kernel pair of some map ft: hom(At,C) —> Xt. This pair is then also the kernel pair of its coequalizer which is, by Proposition 5, the map hom(At,c) where c: C —> Q is a coequalizer of (p,q}. If p',q''- L —> C is c's kernel pair in CoalgS, there exists a unique h: K —» L with p'h = p and q'h = q. For each t e T2hom(At,/i): hom(At,K) —> hom(At,L) is a bijection (as the canonical map between two kernel pairs of hom(At,c)). Thus h is an isomorphism in CoalgS by Proposition 4 and (p, q) is a kernel pair. D

COALGEBRAS WHICH ARE ALGEBRAS

231

7 Proposition hom(A t ,— ) preserves regular epimorphims for each t € T£. The family (hom(At, — )teTs collectively reflects regular epimorphisms. Proof Preservation follows from Proposition 5. Let now q: L —> Q be a homomorphism such that, for each t 6 TS, hom(At,g) is surjective. Let r, s : K —> L be P its coequalizer. Then hom(At, Q is an isomorphism since, for each t 6 TS, hom(A ( ,ft) is bijective. D Corollary In CoalgE, every epimorphism is regular. Proof If e is an epimorphism, so is hom(A t , e) (since hom(A t , e) preserves colimits) for each t € T£. Hence hom(A(,e) is a regular epimorphism for each t £ TE, and so is e by 7. D Corollary At zs (regularly) protective, for each t G IE . Using Propositions 4 to 7 we now conclude by Bunge's characterization of presheaf categories (see [2, 4]) 8 Theorem Let A be the full subcategory of CoalgS spanned by all tree coalgebras. Then CoalgS~Set A ° P . 9 The description of CoalgS as a many-sorted variety of unary algebras as presented in the theorem above, can be simplified by means of the following result. Proposition Every homomorphism /: As —> At has a unique decomposition into embeddings of maximal subtrees. Proof Observe first that, given a tree t and some w e Deft, the map v H-> wv is a homomorphism

tw obviously sends the root of t(w—) to w. Given any homomorphism /: As —* A t , put w — /(e). Since the family (A t ) t is multifree on one generator we conclude s = t(w—) and / = tw. Thus, the only homomorphisms between tree coalgebras are embeddings of subtrees.

232

PORST AND DZIERZON

If w €. Deft is decomposed as w = uv the embedding tw decomposes as A^V^'A^-^A,. Thus, if w = ii... ik with ij £ u>, we obtain a decomposition of tw as Atjij...^-) —> At^!...^..!-) ->

> A t ( i l i 2 _) —^» A t ( i l _) -^-> At

with /e = t(ii...i e _i-)t < ! . This is a decomposition into embeddings of maximal subtrees; it is unique since the decomposition of words into letters is unique. D We denote by QE the following many-sorted signature of unary algebras: • Sorts are all t £ TD; • Operational symbols are f$: t —> t(i—) for all embeddings of maximal subtrees ti. Then, clearly, one has Theorem There is an equivalence of categories Set

A °P

~ AlgQs-

10 We can describe the resulting equivalence CoalgE ~ Algfi£ directly. First, a straightforward calculation gives the following lemma: Lemma For all trees t and for all coalgebras C there is a bijection hom(A t , C) ~ {c e C | tc = t} =r1[i] Let now C = (C,ac) be a S-coalgebra. We define an Qs-algebra XC = ((Ct) t , (if)) by Ct = {c e C | tc = t}, and for c e Ct:

(1)

*f(c) = c i ^=>.ac(c) = ( ( c i ) . . . , c n ) , t(c)).

(2)

If now, /: C —> D is a homomorphism, note first that c e Ct implies /(c) e A (clearly / maps \-l[t] C C into r1^] C £>). In order to prove that the resulting family of maps ft'- Ct —> Dt is an fis-homomorphism it remains to show that, for each ti, the following diagram commutes.

COALGEBRAS WHICH ARE ALGEBRAS

ct

233

Ch

Here Cj is determined by (see equation (2)) otc(c) = while /(c)j is determined by

Since / is a coalgebra homomorphism, we have

thus, /(c)i = /(cj) as required. Denote the functor CoalgS —> AlgQs just defined by $. Next we construct a functor * : Algfig —> CoalgE. Given X = ( ( X t ) , (
C= ]]_Xt and, for x e X t , ters QX(X) = ((i*(a:), . . . , £(*)), t(c)) e FEC.

(3) (4) e want to

If (ft): X —> Y is an rig-homomorphism put / = lit /*• ^ that the following diagram commutes:

Uft

Choose x e Ct. Then JJ/tW = MX) e Kt and

and, also (by equation (4)),

show

234

PORST AND DZIERZON

Since (ft)t is a homomorphism, we have

for i = 1, . . . , n as to be shown. Now we will show, as expected, that \& o $ = id and $ o ty = id. For the former take a coalgebra C = (C, «c) and calculate, with notations as above,

* o $(C) = *(Xc) = CXc - (U Ct, a Note that \[Ct = C and, for c & Ct,

(2)

«=>

Thus, * o $(C) = CXc = C. Obviously, * o $(/) = / for homomorphisms /• In order to prove <J> o $ = id, take an fig-algebra X = (pC t ) teTs ,(*7k e n s ). One gets - XCx = ((C t ) t€Tc , (^ x )f i We start showing that, for each t 6 TS,

To prove Xt C Ct we need to show that, for each x 6 Xt, the tree tx generated by x in Cx equals t. In fact, take w & Defi n Defi x ; it first follows by induction that xw G Xt(w-)'. Clearly xe = x e Xt. For w = vk with k < m and a;,; G -X't(u-) one nasi ^Y definition of ax and tx,

hence

——-x For

these

w

we

obtain

tx(w)

=

t(w)

since

shows: Furthermore, both domains of definition coincide, as induction v Clearly c G Defi n Defi x , and for v with a%(xv) = ((t(v-)t (xv))i<m,t(v))

COALGEBRAS WHICH ARE ALGEBRAS

235

we have w = vk £ Deft x o w e Deftz, k < m by def. of tx <4> v 6 Deft, k < m

by ind. hyp.

4=> vk e Deft

by def. of trees.

Hence i = ix as required. Moreover, for c £ Ct C C there is some s € TE with c € Xa c Cs, hence t = tc = s. Now Xt = Ct follows. Finally, for c 6 Ct = Xt, one has for each U ^(c} = Ci ^ ax(c) = ((c 1 ,...c n ) ) t(e)) 4^ f (c) = Ci. Now $ o $(X) = X. follows. Again, $ o $(/) = / for homomorphisms / is obvious0. References [1] J. Adamek and H.-E. Porst, On Tree Coalgebras and Coalgebra Presentations, Theoret. Comput. Sci. 311 (2004), 257-283. [2] M. Bunge, Relative functor categories and categories of algebras, J. Algebra 11 (1969), 64-101. [3] Chr. Dzierzon and Chr. Schubert, Terminal coalgebras and tree-structures, Preprint, Univ. Bremen, (2003). [4] F.E.J. Linton, Applied functorial semantics II, Lecture Notes in Math. 80, Springer (1969), 53-74. [5] J. Worrel, A Note on Coalgebras and Presheaves, Electron. Notes in Theoret. Comput. Sci. 65, No. 1, (2002), 7 p.

°Note that we didn't prove $ o vp = id literally, since we were tacitly assuming the underlying sets of an algebra to be mutually disjoint; thus <E> o \P and id are, strictly speaking, only naturally isomorphic.

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A HYPERSPACE COMPLETION FOR SEMIUNIFORM CONVERGENCE SPACES AND RELATED HYPERSPACE STRUCTURES

GERHARD PREUSS Institut fur Mathematik I Freie Universitat Berlin Arnimallee 3 14195 Berlin, Germany E-mail: [email protected] Since hyperspaces of complete (separated) uniform spaces are not complete in general, it is highly remarkable that in the more general context of semiuniform convergence spaces even a hyperspace completion exists which preserves several invariants, e.g. precompactness (and thus compactness), connectedness (and uniform connectedness), the property of being a filter space (or a semiuniform space), etc. This completion is used to characterize the subspaces of the compact spaces in the realm of semiuniform convergence spaces axiomatically. The complete hyperspace structure is coarser than the usual uniform hyperspace structure in case uniform spaces are considered. Mathematics Subject Classifications (2000): 54A05, 54B20, 54D30, 54E15, 54E52 Keywords: Hyperspaces, semiuniform convergence spaces, filter spaces, semiuniform spaces, Hausdorff metric, precompactness and compactness, one-point completions and generalizations.

0. Introduction

If X is a space (e.g. a metric space, a topological space, a uniform space or a semiuniform convergence space), then a hyperspace of X, denoted by H(X), is a space whose points are suitable subsets of X and in which X can be embedded (possibly under additional assumptions). The embedding of X into # (X) is said to preserve (resp. reflect) some property P if H(X.) (resp. X) has P whenever X (resp. H(X.)) has P. Hyperspaces of metric spaces have been studied first by F. Hausdorff [3; pp. 290-294] whose distance on the set of all non-empty, closed and bounded subsets of a metric space is known today as Hausdorff metric. Later on, L. Vietoris [8] defined a 237

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topology on the set of all non-empty closed subsets of a topological space X, which is called now the Vietoris topology. In this way one obtains a topological hyperspace provided that X is a topological TI—space. It has been studied extensively by E. Michael [5]. Hyperspaces of (separated) uniform spaces have been considered first by N. Bourbaki [1; p.97, ex.7)] and later on partially by E. Michael [5] and in some detail by J. Isbell [4]. Though the hyperspace of a complete metric space is a complete metric space, a corresponding result for separated uniform spaces is not true. In this article we will prove that the situation becomes much better, when we study semiuniform convergence spaces, which include uniform spaces as well as (symmetric) topological spaces and have many convenient properties (cf. [6]). In particular, if A denotes an arbitrary set of non-empty subsets of a semiuniform convergence space X containing all singletons, there is a coarsest semiuniform convergence structure J7jj on A such that X is a subspace of (.4, JjJ). It turns out that (A, jfy is already a completion of X, whenever A contains a subset of X with at least two points. This completion generalizes the construction of one-point extensions (cf. [6]) for semiuniform convergence spaces with more than one point. The embedding from X into (A, J^) preserves and reflects the following invariants for semiuniform convergence spaces: (1) (2) (3) (4) (5)

precompact, semiuniform limit space, diagonal, semi-pseudouniform, and semiuniform.

Furthermore, it preserves compactness and connectedness as well as uniform connectedness. Last not least, the hyperspace completion (.4, J7jj) is used in order to characterize the subspaces of compact spaces in the realm of semiuniform convergence spaces. Finally, the hyperspace for diagonal uniform limit spaces (— Cook Fischer spaces), introduced by R. J. Gazik [2], is generalized to diagonal semiuniform limit spaces. The embedding from a diagonal semiuniform limit space X into this hyperspace preserves and reflects the following properties: (1) Cook Fischer space, (2) semiuniform, and (3) uniform.

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239

For uniform spaces the original construction mentioned above is reobtained. The terminology of this paper corresponds to [6]. Notation. Unless otherwise stated, in the following, let A be a set of non-empty subsets of a semiuniform convergence space (X, Jx] such that {x} e A for each x € X. If an injective map i : X —> A is defined by i(x) = {x} for each x £ X, we put i[X] = X' and assume without loss of generality that X — X', i.e. i is an inclusion map. 1. Hyperspace structures for all semiuniform convergence spaces Proposition 1.1. If JA = {H e F(A x A): there is some F e Jx with (i x i)(F) C H or there is an ultrafilter U on A having no trace on X such that U x U C H}, then (A, JA) is a hyperspace of (X,Jx) such that the inclusion map i : (X,JX) —> (A, JA) preserves (and reflects) precompactness. Proof. 1) (A,JA) is a semiuniform convergence space: UCl) For each x 6 X,i(x) x i(x) — i(x) x i(x) = i x i(x x x} € JA, and for each A £ A\X,A x A £ JA, since A is an ultrafilter having no trace on X. UC2) is fulfilled by definition of JA. UC3) Let H € JA. Thenl. U D ixi(jF) for some T € Jx or 2. H^UxU for some ultrafilter U on A having no trace on X. In the first case, H'1 D ((i x tJGF))- 1 = (t x iX-F-1), which implies 7T1 e J_J, since F~l 6 Jx- In the second case, H'1 D (U x U)-1 =UxU, i.e. U~l € JSA. 2) (X, Jx) is a subspace of (A, J£), i.e. Jx = (JSA}X = {? & F(X x A")-: (i x i)(?) e JJ}: a) Jx C (JJ)x follows directly from the definition of JA. b) Let T 6 J^)x- Then (i x i)(jF) 6 JA, i.e. 1. (i x i)(F) D (ix i)(Q) for some Q £ Jx or 2. (i x i)(^r) D U x W for some ultrafilter W on .4 having no trace on X . In the first case, (i x i)-1(i x i)(F) = F D (i x i)~ 1 (i x i)(^) = Q , which implies T e Jx- The second case cannot occur, since otherwise U would have a trace on X. 3) (X,Jx) is precompact iff (.A, J"jD is precompact: "<=". Since precompactness is hereditary, this implication follows from 2). "=>•". Let W be an ultrafilter on A . Then 1. i-1(W) does not exist or 2. W has a trace on X. In the first case, U xll & JA by definition, i.e. U

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is a Cauchy filter. In the second case, i~l(U} is an ultrafilter because U is an ultrafilter. Furthermore, X € U (otherwise A \ X G U and U would not have a trace on X). Since (X,JX) is precompact, i~l(U) is a Cauchy filter, i.e. i~l(U) x i~l(U) = (i x i)"1^ x W) belongs to Jx, and thus (note X xX £UxU),(ix i)(i x i)~l(U x U) = U x U belongs to JA by definition, i.e. U is a Cauchy filter. D

Definition 1.2. (.4, JA) is called the simple hyperspace of (X, J7x)Remark 1.3. Let (JA)jej be the family of all semiuniform convergence structures on A inducing Jx, i.e. (JA)x = Jx for each j e J. Then the final SUConv-structure JA on A w.r.t. (1^ : (A,JA) —> .A)je./, where 1^ = I.A for each j e J, i.e. supj6i7v7^ w.r.t. <, is the finest semiuniform convergence structure on A which is coarser than each JA. Obviously, JA = U JA. (A, JA) is called the final hyperspace of (X, Jx). Proposition 1.4. Let JA be a semiuniform convergence structure on A. Then the following are equivalent: (1) JA i-s the final hyperspace structure JA on A. (2) JA is the coarsest semiuniform convergence structure on A such that (X,Jx) is a subspace of (A, JA)(3) JA = {H e F(A x A) : (i x i)~ 1 (W) exists and belongs to Jx or (i x i) l ( H ) does not exist, Proof. "(!)=> (2)". 1. (X,JX) is a subspace of (A,JA), i.e. Jx = (JA)X: If F € Jx, then (i x i)(.F) e JA C JA = JA, i.e. T 6 (J^)x- Conversely, let F 6 (J!A)x- Then (i x i)(.F) e J"i = U ^"i- Hence, there is some jeJ j e J such that (i x i)(^) G jj, i.e. J=" e ( jj) x = Jx2. Let JA be a semiuniform convergence structure on A inducing Jx. Then there is some j € J with JA - JA C JA - JA"(2) =>• (3)" . A semiuniform convergence structure JT^ on ^t is defined by H e JA iff (i x i)~ 1 (W) exists and belongs to Jx or (i x i)"1^) does not exist. Then

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241

1. (X, Jx} is a subspace of (.4, JA): a) If F e Jx, then the filter (i x i}(F) on A x A has the filter J- as a trace on X x X, i.e. (i x i)(F) £ JT^ or equivalently, b) If 7" 6 (J'A)x, i.e. (i exists and belongs to Jx • 2. Let JA be a semiuniform convergence structure on A inducing Jx, and let F e J.A. Then 1° (i x i)"1^) does not exist, i.e. T 6 J^, or 2° (i x i)"1^) exists, which implies that (i x i)(i x i)"1^) D J" belongs to J^, i.e. (i x i)"1(^r) e (J^)x = Jx or equivalently, J- e JA. Thus, J^ C J^. It follows from 1. and 2. that JT"^ is the coarsest semiuniform convergence structure on A inducing Jx. By assumption, J^ = JA"(3) =£> (1)". JA as defined under "(2) => (3)" is a semiuniform convergence structure on A inducing Jx. Thus, JA C (J J7^ = JT^ (cf. Rem. 1.3). i£J

It has been proved under "(1) => (2)" that JA is also a semiuniform convergence structure inducing Jx. Consequently, by 2. under "(2) =*> (3)", 3 A c 3 A- Hence, JA = JA is thefinalhyperspace structure JA. D Corollary 1.5. JA < JA, i.e. lx : (X,JA) continuous.

-» (X,j{) is uniformly

Corollary 1.6. A semiuniform convergence space is precompact iff its final hyperspace is precompact. Proof. "=>". This follows immediately from Prop. 1.1 and Cor. 1.5 "<£=" . Precompactness is hereditary.

D

Remark 1.7. A filter H on A converges to A e A (w.r.t. q-, , ) iff (H n J

A

A) x (H n A) e JA. Thus, using Prop. 1.4 (3), "V

1. W —4 A e .4 \ X iff W € 7^/ , i.e. U is a Jj-Cauchy filter, 9

V

2. H —4 x e X i f f i-l(Hr\x) This means that for each Ti. € f (.4), we obtain: If i~1(Ti) exists, then

242

PREUSS T

a) for each x € X, H -^ x iff r J (W) -^ a; «V b) for each A e ,4 \ X, W — 4 A iff t-^W) 6 jjx (i.e. W 6 and if i 1(7i) does not exist, then for each A e A,H —4 A. In particular, a filter H on .4 converging to some x £ X, converges to each AtA\X. Theorem 1.8. For each semiuniform convergence space (X,J~x), the final hyperspace (A,J^fy is complete whenever A \ X is non-empty, and it contains X as a dense subset, i.e. it is a completion o f ( X , J x ) Proof. Each J^— Cauchy filter H on A converges to each A & A \ X (cf. Rem. 1.7). Furthermore, each A e A is an adherent point of X: For each x 6 X, i(x) converges to x and contains X, i.e. x is an adherent point of X, and it converges to each A € A \ X, i.e. each A € A \ X is an adherent point of X. D Corollary 1.9. Let (X, 7) be a filter space, (X,^y) its corresponding semiuniform convergence space, and (A,Jj) the final hyperspace of (X,J~/), where A \ X is non-empty. Then the underlying filter space ( A , ^ j f ) is complete (i.e. a symmetric Kent convergence space) and contains (X, 7) as a (dense) subspace. In particular, 7-7 is the coarsest filter space structure on A inducing 7 . Proof. Since (X, J~f) is a subspace of (.4, jfy, (X, 7^ ) = (X , 7) is a subspace of (A,jjf) (cf. [6; 2.3.3.17]). Obviously, -yjf = {7" € F(A) : i~l(F) does not exist or (i~l(F) exists and belongs to 7)}, and by Thm. 1.8, (A,ijt) is complete and X is dense in (A7j/)> i-e- in (A
A

J

A

D

Corollary 1.10. Let (X,q) be a symmetric Kent convergence space, (X,~fq) its corresponding filter space and (A,J^^) the final hyperspace of (X,J^q). Then the underlying (symmetric) Kent convergence space (A,q~, f ) of(A,jJi) contains (X,q) as a (dense) subspace.

HYPERSPACE COMPLETIONS

243

Proof. Since (X, J^g) is a dense subspace of (A, JjJ), (X, q) is a dense subspace of (A, q~, ,). D

Remark 1.11. (1) If A contains all singletons and exactly one A with cardinality at least 2, then (A, j f y is the one-point extension of the semiuniform convergence space (X,Jx) (cf. [6; 3.2.4]), and it is called the onepoint completion o f ( X , J x ) (2) Since subspaces of convergence spaces (=syrnmetric Kent convergence spaces) are filter spaces, it follows from Cor. 1.9 that the filter spaces are exactly the [dense] subspaces (formed in the construct SUConv of semiuniform convergence spaces) of the convergence spaces - a result which is already known (cf. [6; 4.4.7]). Theorem 1.12. Let (X,Jx) the following are equivalent:

be a semiuniform convergence space. Then

(1) (X,Jx) is precompact. (2) (X,Jx) is a dense subspace (in SUConv) of a compact semiuniform convergence space. (3) (X, Jx) is a subspace (in SUConv) of a compact semiuniform convergence space. Proof. "(1) => (2)". If (X, Jx) is a semiuniform space, then 1. \X\ < 1 or 2. \X\ > 1. In case 1, the assertion is obvious, since (X, Jx) is already compact. In case 2, let A contain all singletons and a subset A of X with at least two points: Then (A, j f y is a completion of (X, Jx), which is precompact by Cor. 1.6. Consequently, (A, J^) is a compact semiuniform convergence space containing (X, Jx) as a dense subspace. "(2) => (3)". This implication is obvious. "(3) =>• (1)". Subspaces of compact semiuniform convergence spaces are precompact, since "compact" implies "precompact", and precompactness is hereditary. D

Proposition 1.13. // (X,Jx) is a compact semiuniform convergence space, then its final hyperspace (A, j f y is compact.

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Proof. Let |X] < 1. Then the assertion is obvious. If |X| > 1, then the assertion follows from Cor. 1.6 and Thm. 1.8 provided that A \ X =£ 0. In case A \ X = 0 and X\>l,(A, J^) = (X, Jx) is compact. D

Proposition 1.14. Let 8 be a class of semiuniform convergence spaces consisting of T\ —spaces. If (X, Jx) is an £—connected semiuniform convergence space, then its final hyperspace (A, J^) is £—connected. Proof. Since X is dense in (A, J^), the result follows from (cf. [6; 5.1.20]). D

Remark 1.15. Let (X, V) be a separated uniform space in the sense of A. Weil [9]. In case A is the set of all non-empty closed subsets, then for each V e V, let H(V) = {(A, B)tAxA:Ac V[B] and B C V[A}}. Obviously, {H(V) : V £ V} is a base for a uniformity H(V) on A, which has been introduced by N. Bourbaki [1 ; p. 97]. The following are valid: (1) (A", V) is a subspace of (A, H(V)). (2) (A, H(V)) is separated (use that for each A&A,A = f| V(A\). vev (A, H(V)) is called the uniform hyperspace. In contrast to the situation for the final hyperspace of a semiuniform convergence space, a) X is not dense in (A, H(V)) and b) (A, H(V)) is not complete, in general: a) Let (X, V) be the usual uniform space Ru of real numbers and A the closed unit interval [0,1]. Then A e A is not an adherent point of X: Choose V = { ( x , y ) e R x R : | x - y \< ||. Then H(V)(A) = {B e A : (A,B) e H(V)} is a neighborhood of A which does not contain any point of X , since there is no x £ R such that A C V(x) = {y e X : (x, y) e V}. b) Even if (X, V) is complete, (A, H(V)) need not be complete (cf. [10; 39.D.2.] or [4; 11.50]). On the other hand, the uniform hyperspace of a (completely) metrizable semiuniform convergence space [= (completely) metrizable uniform space] is (completely) metrizable (cf. [10; 36.E.2. and 39.D.I.]), where the metrizability is realized by means of the HausdorfT metric. J. Isbell [4] uses (separated) uniform spaces in the sense of W. J. Tukey [7]

HYPERSPACE COMPLETIONS

245

and defines on the set A of non-empty closed subsets of a separated uniform space (X, /z) (described by means of uniform covers) a uniformity as follows: Put H(U) = {(A,B) & Ax A: AC St(B,U} and B C St(A,U}} for each U € ^. Then (H(U) : U £ /z} is a base for a separated uniformity in the sense of Weil. If (X, V) is a separated uniform space in the sense of Weil and (X, ^v) its corresponding uniform space in the sense of Tukey, then it is easily checked that H(V) has the base (H(U) :U £ fJ-v}, i.e. Bourbaki's and Isbell's definition of the uniform hyperspace coincide. If (X, V) is a separated uniform space and (X, [V]) its corresponding semiuniform convergence space, i.e. [V] = {F € F(X x X) : F D V}, then (,4, [H(V]) is a separated semiuniform convergence space containing (X, [V]) as a subspace. Therefore, [H(V)] is finer than J^, i.e. [H(V)} C J^.

2. Hyperspace structures for semiuniform limit spaces Proposition 2.1. A semiuniform convergence space (X,Jx) niform limit space iff (A, jTjJ) is a semiuniform limit space.

is a semiu-

Proof. "•<=" . Since the construct SULim of semiuniform limit spaces is bireflective in SUConv (cf. [6; 2.3.2.3]), this implication is obvious. "=>". Let T,Q £ j{. Then 1. (i x i)~l(F) exists and belongs to Jx or 2. (i x i)~l(F) does not exist, and 3. (i x i)~l(Q) exists and belongs to Jx or 4. (i x i)~l(Q) does not exist. If "1. and 3." is valid, (i x i)~l(FnG) = (i x i}~l(F) n (i x i)~l(G] belongs to Jx by assumption, i.e. J"n Q e JjJ. In case "1. and 4." is true, (i x ^(.Fn <J) = (i x i)"1^) 6 v7x, i.e. ^"n 5 e J_4. Similarly, one concludes in case "2. and 3." is true. If "2. and 4." is satisfied, (i x ^"-^.Fn £7) does not exist, i.e. T T\ Q e JjJ. D Corollary 2.2. Let (X,J~x) be a semiuniform limit space. Then (A,J^} is the coarsest semiuniform limit space structure on A inducing Jx • Proposition 2.3. A semiuniform convergence space (X,Jx] pseudo-uniform iff (A, J^) is semi-pseudouniform.

is semi-

Proof. "«=". By [6; 6.3.2.3) b)], SPsU is bireflective in SUConv. "=$>". If H is a filter on A x A such that H <£ j£, then (i x i)~1(H) exists and does not belong to Jx • This implies that there is an ultrafilter U on X with U D (i x i}~l(H) and U £ Jx. Consequently, i x i (U) D H is an

246

PREUSS

ultrafilter which does not belong to jj^, since (i x i)~l(i x i)(U) — U^. JxThus, the assertion is proved. D

Remark 2.4. The inclusion map i: (X, Jx) —» (-4, JjJ) does not preserve the properties "uniform limit space" and "uniform space", e.g. if C denotes one of the constructs ULim or Unif, there is no coarsest C—structure £ on {0,1,2} such that ({0,1,2},£) contains the discrete uniform space D% with underlying set {0,1} as a subspace (cf. [6; 3.2.7.©]). Proposition 2.5. A semiuniform convergence space (X,Jx) is semiuniform (=principal semiuniform limit space) iff (A,J^) is semiuniform. Proof. "<=". The construct SUnif of semiuniform spaces is bireflective in SUConv (cf. [6; 6.3.2]). "=$>". By Prop. 2.1, (A,J^) is a semiuniform limit space. In order to prove that (A,J^) is principal, it must be verified that f) H € J^' Since Jx — [Q], i.e. there is some Q £ Jx such that each f e Jx contains Q, and the trace of |") Ti. on X x X exists, one obtains (ixi)-l(

ft

W) = (ixi)- 1 (n{W'ej{: ( i x i ) - l C H f ) exists and belongs

to Jx} = r\{(i x O'HW) = W 6 JA and (i x i)-l(H'} e Jx} D a, i.e. (i x i)- J ( H W) £ Jx which implies f) W e JjJ.

Proposition 2.6. yl semiuniform convergence space (X,Jx) iff (A, J A) is diagonal.

is diagonal

Proof. "4=". Since, obviously, the construct A -SUConv of diagonal semiuniform convergence spaces is bireflective in SUConv, this implication is true. «=>". Since (i x i)-1^)) - (A x ) £ Jx, (A^) € jj. D Remark 2.7. It follows from the above results that for each diagonal semiuniform limit space (X, Jx) the hyperspace (A, jfy is a diagonal semiuniform limit space. In order to obtain a hyperspace (A, Jj?) of (X, Jx) such that the inclusion map from (X, Jx) into (.4, J f ) preserves the properties "diagonal uniform limit space" and "uniform" some further definitions are

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247

necessary in the following. Definition 2.8. 1) Let (X, Jx) be a semiuniform convergence space. A non-empty set B C Jx is called a base for Jx provided that each F 6 Jx contains some Ti. 6 B. 2) Let (X, Jx) be a diagonal semiuniform limit space. Then Bc = {? n F~l n (Ax) : T € Jx} is called the canonical base for JxRemark 2.9. 1) Each filter H £ Bc has a base consisting of symmetric elements containing the diagonal. In particular, each Ji € Bc contains the diagonal. Obviously, Bc = {H € Jx • H = U~l and H C (Ax)}. 2) If (X, Jx) is a diagonal semiuniform limit space, then for each H £ Bc, a filter T(Ti) on A x A is defined which is generated by the filter base {T(-H,H) : H e H}, where T(U,H) = {(A,B) e A x A : A C Jf[B] and BC Theorem 2.10. Let (X, Jx) be a diagonal semiuniform limit space. Then JA* = {K. e F(A x A) : K, D T(ft) /or some H e £c} is a diagonal semiuniform limit space structure on A inducing Jx • Proof.

(1) {T(H, #) : if e K} is a filter base for each H € Bc: a. T(W,#) ^ ^, since A^ c T(H,H). b. If Hi, #2 € W, then T(W, FI n F2) C T(W, FI) n (2)

(3)

a) (A^) £ JA* follows from T((A X )) C (A^). b) JA* is closed under formation of superfilters. c) K. € JjJ implies that there is some W e Bc such that /C D T(W).Consequently, K'1 D (T(W))-1 = T(H) (note: T(W,H) = (r(W,F))- 1 for each H € W), i.e. /C"1 6 J^*. d) Let /Ci,/C 2 € J&*. Then ACi D TXfti) and ^2 3 T(H^) withWi,W 2 £ Bc. Hence, T(Hi n H 2 ) C T(Wi) n T(W2) C ^i n K.2 which implies /Ci n /C2 6 ^*, since HI n W2 e Bc. a) For each H e Bc, (i x i)- J (T(H)) = H: By Rem. 2.9. 1), 7i has a base BU consisting of symmetric elements of H. (which contain the diagonal). Then {T(H, -Bw) : BH e B-H] is a base of T(H). Furthermore, T(H, BH] n (X x X) = B-H, since 5H is symmetric. Thus, (i x i)"1^^)) and W have the same base 6^1 i-e. a) is proved.

248

PREUSS

b) JA* induces Jx, i.e. (X, Jx] is a subspace (in SUConv) of (A,JA*):

a) Let F & Jx- Then there is some W 6 Bc such that JF D H. Using a), (ixi)(F) D (txi)(W) = (ixzX^xi)- 1 ^^))) 3 T(W), i.e., (i x t)(.F) € &*. /?) Let J" 6 F(X x X) such that (i x i)(f) e J&*, i.e. there is some H € Bc with (i x i)(JF) D T(U). Hence, J" = (i x i)~l((i x i)(^)) D (i x i)-1^^)) = H, which implies ^€JX. D Corollary 2.11. Let (X, Jx} be a diagonal semiuniform limit space. Then JA*CJA, i.e. JA* is finer than JA. Corollary 2.12. A semiuniform convergence space (X,Jx) is a CookFischer space (=diagonal uniform limit space) iff (A, JA*) is a Cook-Fischer space. Proof. "<=". Since the property "Cook- Fischer space" is hereditary, this implication is obvious. "=»" . Let K.I , /C2 e Jjf such that /Ci o/C2 exists. Then there are U\ , Hz 6 Bc withT(Wi) C /Ci for each i e {1,2}. In particular, T(Hi)oT(Ui} C /Cio/C 2 , and because of T((W2 o Wi) n (H\ o W 2 )) C T(Wi) o T(W2) the assertion follows since (W2 o W : ) n (Wi o W 2 ) g Sc (note that the composition of elements of Bc always exists). D

Corollary 2.13. A semiuniform convergence space (X,Jx) form (uniform) iff(A,jA*) is semiuniform (uniform).

is semiuni-

Proof. "<=" . The property "semiuniform" ( "uniform" ) is hereditary. "=>". a) If (X, Jx) is semiuniform, i.e. a principal semiuniform limit space, then (X, Jx) is diagonal. In order to prove that (A,J^*) is a principal (diagonal) semiuniform limit space, it suffices, by Thm. 2.10, to verify that Jjf is generated by a single element of Jj*. By assumption, there is some H £ Jx such that Jx = [W]- Since Ji= fj ? e Jx is a semiuniformity, U e Bc, and [T(7i)} = 3f (namely, if K, 6 &*, there is some H' £ Bc C Jx with /C D T(W'); hence- W C W and T(H) C T(W), i-e. /C e [T(H]}). b) If (X, Jx) is uniform, i.e. a principal uniform limit space, then (X, Jx) is diagonal, and by Cor. 2.12, (A,ffi) is a diagonal uniform limit space. Thus by a), (A, JA*) is principal, i.e. a uniform space. D

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249

Remark 2.14. If (X, Jx) is a separated Cook-Fischer space and A is the set of all non-empty compact subsets of (X, Jx), then (A,Jj?) is the hyperspace constructed by R. J. Gazik [2]. In case (X,JX) is a separated uniform space, i.e. Jx = [V] where V is a separated uniformity on X, and A is the set of all non-empty closed subsets of (X, Jx), then Jjf — [T(V)J, and (A,T(V)) is Bourbaki's uniform hyperspace, i.e. T(V) = H(V) (cf. Rem. 1.15). References [1] Bourbaki, N.: 1940, Topologie generate, Chapitres 1 et 2, Hermann, Paris [2] Gazik, R. J.: 1974, 'Uniform convergence for a hyperspace', Proc. Amer. Math. Soc. 42, 302-306 [3] Hausdorff, F.: 1914, Grundziige der Mengenlehre, Veit, Leipzig. (Reprint: 1949, 1965 Chelsea, New York; 2002 Springer, Berlin.) [4] Isbell, J. R.: 1964, Uniform spaces, Math. Surveys No. 12, Amer. Math. Soc., Providence. [5] Michael, E.: 1951, 'Topologies on spaces of subsets', Trans. Amer. Math. Soc. 42, 152-182. [6] Preufi, G.: 2002, Foundations of Topology, Kluwer, Dordrecht. [7] Tukey, W. J.: 1940, Convergence and Uniformity in Topology, Ann. of Math. Studies 2, Princeton University Press, Princeton. [8] Vietoris, L.: 1922, 'Bereiche zweiter Ordnung', Monatsh. fiir Math, und Phys. 32, 258-280. [9] Weil, A.: 1937, Sur les espaces a structures uniformes et sur la topologie generate, Hermann, Paris. [10] Willard, S.: 1970, General Topology, Addison-Wesley, Reading.

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CONVEX EFFECT ALGEBRAS AND PARTIALLY ORDERED POSITIVELY CONVEX MODULES

D. PUMPLUN Fachbereich Mathematik FernUniversitat 58084 Hagen Germany E-mail: dieter.pumpluen@>fernuni-hagen. de

HELMUT ROHRL 9322 La Jolla Farms Rd. La Jolla, CA 92031 USA E-mail: [email protected] Convex effect algebras have been introduced by S. Gudder and S. Pulmannova in connection with quantum mechanics, enriching the notion of an effect algebra. It is shown that convex effect algebras can be equivalently described as partially ordered, preseparated positively convex modules satisfying an order convexity condition and being equipped with an affine automorphism of order 2 satisfying a characteristic equation. Mathematics Subject Classifications (2000): 81R10, 52A01, 52A05 Keywords: positively convex module, convex effect algebra, order, quantum mechanics

1. Introduction

Effect algebras have been studied in connection with quantum physics and fuzzy probability theory. As most effect algebras, that appear in applications, have an additional convex structure, S. Gudder and S. Pulmannova introduced convex effect algebras and investigated them together with S. Bugajski and E. Beltrametti (cf. [2], [3], [4]). 1.1.

Definition

([2], [3]): An effect algebra is a quadruple (A, ©,0,1) consisting of a nonempty set A, two elements 0,1 6 A and a partial binary operation 0 251

252

PUMPLUN AND ROHRL

satisfying the following conditions: (El) If, for a, b £ A, a © b is defined, then so is b © a and a © b = b © a holds. (E2) If, for a, 6, c e A, a © b and (a © 6) © c are defined then 6 © c and a © (6 © c) are defined and (a © b) © c = a © (6 © c) holds. (E3) For every a £ A there exists a unique a' e A with a © a' = 1. (E4) If a © 1 is defined for an a € A then a = 0 follows. In the following an effect algebra will be simply denoted by A if misunderstandings are not possible. An effect algebra A will be called a convex effect algebra, if, for every a € A and every A € R with 0 < A < 1, there is a Aa 6 A such that the following conditions are satisfied: (Cl) For any a, /3 £ R with 0 < a, /3 < 1 and any a e A, (aj3)a = a((3a) follows. (C2) For any a, /? > 0 with a + /3 < 1 and any a e A, (aa) © (/3a) is defined and equals (a + /3)a. (C3) For any a, b € A such that a © b is defined and any A with 0 < A < 1, (Aa) © (A6) is denned and equals A(o © 6). (C4) For any a £ A la = a holds. It is shown in [3] that in a convex effect algebra A, for any a, b € A and any a, (3 > 0 with a + /3 < 1, (aa) © (j3b] is defined. This means that A, n

with the inductively defined sum 0(ajai), for n 6 N, a^ € A, 1 0, 1 < i < n, with £) a, < 1, is a positively convex module (cf. i=l

[5], [8], [11], [12], [13]), because the defining equations for positively convex modules follow easily from the axioms of a convex effect algebra. A convex effect algebra A also posesses a canonical partial order "<" . One defines a < b, a, b £ A, iff there is a c 6 A such that b = a © c (cf. [2], [3]). This partial order is compatible with the positively convex operations in A, i.e. A is a partially ordered positively convex module (cf. [2]; [3], 2.1; 2.3). One would like to characterize those partially ordered positively convex modules which come from convex effect algebras. Hence partially ordered positively convex modules are introduced in §2. In §3 the relation between convex effect algebras and partially ordered positively convex modules is discussed. It turns out that the partial order of a convex effect algebra considered as a positively convex module is a preorder which exists on any positively convex module and was first defined by Wickenhauser [13]

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253

in 1988. As a consequence it is proved that the category of convex effect algebras is equivalent to a category consisting of preseparated, proper positively convex cone modules P possessing an affine endomorphism which fulfills an equation corresponding to the existence of an additive complement (cf. [2], 2., (E3)) in an effect algebra and with positive-order-convex 2~1P (cf. 3.2, 3.4, 3.5). 2. Ordered positively convex modules. Positively convex modules were discussed in [5], [6], [8], [10] and [11] and partially ordered positively convex modules were investigated in [7] by Nortemann. For the reader's convenience the definitions are repeated here. n

ftpc := {a/a = (ai, . . . ,a n ),n € N, en > 0, 1 < i < n, and £) a» < 1} 1=1 denotes the set of positively convex operations.

2.1.

Definition

(cf. [8], [10]): A positively convex module P is a non-empty set together with a family of mappings &p : Pn —> P, a = ( a i , . . . , a n ) € fipc. In addition, with the notation n a

iPi :=ap(pi,...,pn)

for pi e P, 1
(PCI) Pi & P, I < i < n, and Sik the Kronecker symbol, 1 < i, k < n. n

a

/

\

m

/

n

E H E fop* = E E

(PC2)

a

fc=l

where (ai, . . . , a n ), (/3ik \ k £ Ki) € Q pc , 1 < i < n, pk e P, for k € (J i=l

n

Moreover, |J Kt = Nm = {k \ I < k < m} and in the "sum" £) A the summands are supposed to be written in the natural oder of the fc's. A number of computational rules follow from these equations (cf. [8], [9], n

[11]), e.g. the fact that J^ oupi is not changed by adding or omitting

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PUMPLUN AND ROHRL

summands with zero coefficients. Hence, (PC2) takes the more simple form

Obviously, any positively convex set, which will always mean a positively convex subset in some linear space, is a positively convex module. Any positively convex module is a convex module (cp. [10]), the converse does not hold, because any positively convex module contains a zero element n

0 :— £) OP*- However, every absolutely convex module ([9]) is a positively i=l

convex module, as is any cone or convex set in a linear space containing the orgin. A familiar example of positively convex sets are the universal caps of cones in functional analysis ([14]). If PI , PI are positively convex modules a mapping / : P\ —> P2 is called a morphism or a positively affine mapping if

i=i holds for any (QI, ... . ,a n ) € tlpc and any Xi 6 P, 1 R+, support of h finite and £) h(x) < 1}, R+ := {t t 6 R, t > 0}, is the free positively X

convex module generated by X ([8]). U : PC —> Set is algebraic. It is the Eilenberg-Moore category of the category Vecf of regularly ordered normed linear spaces where A : Vecf —> Set is the forgetful functor with A(£) := (x x e E, x > 0, ||x|| < 1} ([5], [6], [8]). A positively convex module will often simply be called a "PC-module" , for the sake of brevity. As any PC-module is also a convex module ([10]) it makes sense to talk about preseparated PC-modules. In [11], 2.4, it is shown that a PC-module is preseparated iff ax + 0z = ay + 0z, with (a,/3) e fipc, a > 0, and x,y,z e P, implies x = y. For PC-modules the following notions also play a role in this paper. 2.2.

Definition:

Let P be a PC-module. (i) P is called weakly preseparated if, for any x,y e P and any 0 < a < 1, ax = ay implies x — y. (ii) P is called proper if, for any x,y e P and any (a, /3) £ fipc with a > 0, ax + f3y = 0 implies x = 0.

ALGEBRAS AND MODULES

2.3.

255

Definition

(cf. [7]): A PC-module P with a preorder "<", i.e. a reflexive and transitive relation "<", is called a preordered PC-module, if "<" is compatible with the positively convex operations, i.e. for any Xi,y, e P, 1 < i < n and any ( a i , . . . , an) S Qpc, Xi < j/j, 1 < i < n, imply n

v~v

n

v^

2_j » « — 2-^i J^*' i=l

<

t=l

In a preordered PC-module P, 0 < a < /3 < I implies for any x € P ax < fix, as the decomposition /3 = a + (f3 — a) and 2.3 together with the computational rules ([8]) show. A preordered PC-module P is called partially ordered if its preorder is also antisymmetric. A positively affine mapping / : PI —> P2 between preordered or partially ordered modules, respectively, is called a morphism of preordered (partially ordered) modules if it is isotone. These morphisms and the preordered morphisms form the category Pr-PC of preordered PCmodules. Preordered PC-modules are sometimes called Pr-PC-modules for short. Analogously the category Po-PC of partially ordered PCmodules and the notion Po-PC-mocfo/es are introduced. It is an easy exercise to show that Po-PC is a full, surjective-reflective subcategory of Pr-PC. For P € Pr-PC and x,y e P one defines x ~ y if and only if x < y and y < x holds. Then R(P) := P/ ~ is the reflection. 2.4.

Definition:

A Pr-PC-module (Po -PC-module) P is called (i) order preseparated, if, for any x, y, z e P and any (a, /3) e fipc with a > 0, ax + /3z < ay + (3z implies x < y. (ii) weakly order preseparated, if, for any x, y 6 P and any 0 < a < 1, ax < ay implies x < y. For a,(3 e R one puts as usual [a,/3] := {x/a < x < /?}, [a,/?[:= [a, ft \ {0}, }a,ft:= [a, ft \ {a}, }a, ft := (a, ft \ [a, /?}. 2.5. Proposition (cp. [1], [7], [8], 3.1; [9], 4.1; [10], 1.2): Let P be a preordered PC-module and define for x,y £ P SXty := {a/0 < a < 1 and there are z e P and

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13 € R with (a, f3) e Qpc and az + /3z < ay + f3z}. Then Sx,y = {0}, [0,1[ or [0,1] holds. Proof: Let Sx 0 and ax + j3z < ay + (3z. If 0 < 7 < a, then (7, a~l^(3) € £lpc holds and by multiplying the above inequality by a-17 one gets jx + a~17/3z < 7j/ + a~17^2, which implies 7 e 5XJ/. Hence, [0, a] c SX1, has been proved. If a = 1, the assertion holds. If 0 < a < 1, using the computational rules for PC-modules (cf. [8], [11]) ax + f3z < ay + f3z implies 2o_ (3 _ a 1 1+a 1+a 1+a 1+a

a

l +a

1+ a

2a (3 < -y v H- -—z. - l+a 1 +a Hence, 2a(l + a)-1 £ Sx,y holds. One defines a\ := a, an+i :— 2a n (l + an)"1 for n e N. (an/n € N) is a strictly increasing sequence which implies lim an = 1, hence [0, l[c Sx>y.

n —^oo

A special type of Pr-PC-module (Po -PC-module) will be important later on: P e Pr-PC (Po -PC) is called a preordered (partially ordered) cone module, if, for any x € P x > 0 holds. Examples for preordered or partially ordered cone modules are order cones in ordered linear spaces, the intersection of the order cone with the unit ball in a Riesz normed linear space (cf. [14]) or a universal cap ([14]). 3. Convex effect algebras and partially ordered cone modules.

3.1.

Definition

(cf. [4], p. 361): For effect algebras A,B, a morphism
B is a mapping with the properties (EM) (!) = 1 and, if ai©(ai)® v?(a2) is defined and equals (p(a\ A morphism of convex effect algebras A, B,
B is a positively afSne morphism of effect algebras (cf. [3]). These morphisms together with

ALGEBRAS AND MODULES

257

the convex effect algebras constitute the category Conv-EfFAlg of convex effect algebras. n

If A is a convex effect algebra, one defines inductively 0 (0^2:,), for any n € N, any ( a i , . . . , a n ] £ fipc and any x^ £ A, 1 2. If ( a i , . . . , a n +i) £ fipc put An := ^ a^ and we may assume n

An > 0. Then Q(aiA~lXi) exists and (An,an+i) £ fipc holds, hence t=i n+l

0(T1^) ®(a n + ii n + i) (3.1.1) 1=1 n is well-defined because of [3], 2.1. A routine computation shows that with this definition (PCI) and (PC2) of 2.1 are satisfied, s.th. a convex effect algebra is in particular a positively convex module. To unify the notation we will write

1=1 »=i If / : A —> S is a morphism in Conv-EfFAlg it is trivially also a morphism in PC if A and B are considered as PC-modules. For A 6 Conv-EfFAlg, denote A considered as a PC-module by M0(A). Hence we have a functor MQ : Conv-EffAlg —* PC. In the following a mapping f '. PI —> P%, Pi € PC, i = 1, 2, is called affine (cf. [10]) if it is a morphism of convex modules, i.e. preserves only the convex combinations: For Xi £ PI and on > 0, 1 < n

i < n, with J2 ai — 1

holds. 3.2. Proposition: For a convex effect algebra A the following statements hold: (i) Mi/i £/ie order "<" on A (cf. §1; [3]) Mo(A) is a partially ordered PC-module denoted by M(A). (ii) M(A) is an order preseparated, proper cone module.

258

P UMPL UN AND R OHRL

(\\i) ^M(A) is positive-order-convex (cf. [14]) i.e. 2~1M(A) = [O^-1!] = [x/x e M(A) and 0 < x < 2"1!}. (iv) The mapping 7 : M(A) —> M(A) defined by -y(x) := x', x & A, and x' its additive complement ([3]), is affine, antitone and a reflection, i.e. satisfies 72 = idA- Moreover, for any x £ M(A)

^x + i7(x) = i7(0) holds. (v) /br an?/ a;, ?/ e M(A), x
I ay — ax + -z. Proof: (i) follows directly from the results in [3] and the definition (3.1.1) of the positively convex structure on Mo(A). (ii): As x = x © 0 holds for any x e A, 0 < x follows. If x,y e M(A), (a, 13) £ fipc, ax + (3y = 0 is the same as (ax) © (f3y) = 0 in A. Now, in A we have 1 = 1©0 = \@(ax)@(Py), hence (ax)® I and (/3x)©l are denned which implies ax = /3y = 0 ([3], (E4)). Because of [8], (3.1), this implies (1 — a)x = 0 if a > 0 and x = (ax) © ((1 — a)x) = 0 follows ([3]). Hence M(A) is proper. Now, let x, y, z € M(yl), (a, /3) € fipc, a > 0 and assume

ax + /3z < ay + 0z. The definition of "<" in [3] implies the existence of a u 6 M(A) with

(ay) @ (0z) = (ay) © (13 z) © u, hence ay = (ax) © it follows (cf. [3]), i.e. ax < ay. As a > 0 holds, 2.5 implies (1 — a)x < (1 — a)y which yields x = (ax) © ((1 - a)x) < (ay) © ((1 - a)y) — y.

(iii): If 0 < x < 2~ 1 a, a,x € M(A), a = (2- J a) © (2- J o) in A ([3]) implies that x © x is denned, hence x = (2~ 1 x) © (2-1x) = 2 -1 (x © x) follows, i.e. M(A) is positive-order-convex. (iv) For x £ A, x' e A is uniquely determined by x © x' = 1 (cf. [3], (E3)). From this both 7(0) = 1 and -x+-7(x) = -

A L GEBRA S AND MOD ULES

259

follow, x < y in M(A) is equivalent to y' < x' ([3]) and (x1)' — x holds, i.e. 7 is antitone and a reflection. For 0 < a < 1 and x,y & M(A), 1.1, (E3), yields

I = (al) © ((1 - a)l) = [a(x © x')} © [(I - a)(y © y')} = (ax) © ((1 - a)y) © (ax') © ((1 - a)y'), hence 7(0:0; + (1 — a}y) = 0:7(0;) + (1 — a)^(y) follows and 7 is affine. (v): Consider x,y € M(A) with x < y. Then there is a u e A with y — x © u which implies 1

1

1

Conversely, ay = ax + 1~lv with 0 < a < 2"1 and v e M(A] means ay = (ax) © (-v)

in A, i.e. ax < ay or x < y because of (ii). 3.2, (v), is particularly remarkable because already in 1988 Wickenhauser introduced in [13] a preorder in every PC-module P by defining x C y, x, y 6 P, if there exists an a with 0 < a < 2"1 and a u £ P such that ay = ax + 2~1u holds. 3.2, (v), shows that the partial order in a convex effect algebra coincides with "d" in the underlying, PC-module MO (^4). All order notions in the following proposition refer to the preorder "C" of PC-modules. 3.3. Proposition: (cp. [13]): Let P, P' be PC-modules, then (i) Any positively affine mapping f : P —> P' is isotone. (ii) P is a preordered weakly preseparated cone module. (iii) // P is preseparated then P is order preseparated. (iv) If P is partially ordered then P is proper. (v) // P is preseparated and proper then P is partially ordered. (vi) Let 7 : P —> P be a mapping, such that -x + i7(z) = 17(0)

holds for any x £ P. Then (a) P = (0,7(0)] and

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PUMPL UN AND ROHRL

(b) if P is preseparated then 7 is an affine, 2~17(0) is the only fixed point 0/7.

antitone reflection and

Proof: (i) is obvious because of the definition of "d" . (ii): As 2~lx = 2~ x + 2~ 1 0 holds, "d" is reflexive and x u 0 holds for any x e P. Let x d y and y d z, x,y,z e P. Then there are u,v £ P and 0 < a, /3 < 2"1 such that l

ay = ax -|— u and /3z = /3y + -t> ,z ,z hold. With 7 := minfa, /?} 7y = 70; + — u and 72 = 7y + — v

follow, which imply

i.e. x d 2. It remains to show that "d" is compatible with positively convex operations. For this, consider Xj d yt, X i , y i € P, 1 < i < n, and ( a i , . . . , a n ) e £lpc- Xi d j/i means Ajy, = AjXj + 2~ 1 u i , with u< e P, and 0 < Aj < 2"1, 1 < i < n. With A0 := min{A, | 1 < i < n} one gets Aoy« = AQX, + Ao(2Aj)~ 1 Uj, 1 < i < n, and hence n - \—v ^0

,

n V—N

-i 1

n

with Vi := (AoAj 1 )uj, 1 < i < n. This means tjXj d If ax d ay with 0 < a < 1 in P, then Aay = Aax + -u &

1

follows with suitable 0 < A < 2" and u G P, hence we have x d y. (iii) Let ax + /3z d ay + /?z, x, y, 2 £ P, (a, /3) 6 flpc and a > 0. Then there are a u € P and a A with 0 < A < 2"1 with Aay + A/3z = Aax + A/3z + -u.

This implies 1

A^ __ 1

1.

A/3

ALGEBRAS AND MODULES

261

and Xay = Xax + 2~lu, because P is preseparated, and we have x E y. (iv): If ax + (3y = 0 in P, for (a, (3) € fipc and a > 0, one gets a 1,8. a 4«+5(i»)=40,

hence 0 Zl a; Zl 0 follows. As "d" is antisymmetric x = 0 follows, i.e. P is proper. (v): Now, let P be preseparated and proper, and consider x C y d x. If one puts z = x in (*) in the proof of (ii)

results. This implies 7 2 o~ 1 u + 72/3~1i> = 0 o r u = u = 0 and, hence, x = y. (vi): The equation for 7 implies x C 7(0) for any x e P, i.e. P = [0,7(0)]. To prove (b) note that 7 is uniquely determined by the equation because P is preseparated. Let x,y € P and a € [0, 1]. Then -(ax + (1 - a)y) + -(aj(x) + (1 - 0)7(2;)) = ^(^

+

2

^ - ^)(-y + ^(y}} = -7(0) holds. As P is preseparated this yields -(ox+(l-o)j/)+-(o7(o;)+(l-o)7(j/)) = -(ox+(l-o)y)+-7( and 7(00; + (1 — a)y) = 07(2;) + (1 — 0)7(7;), i.e. 7 is affine. For any x £ P 1 1 . . 1 . . 1 2. . -x + -7(0;) = -7(x) + -Y(x) implies 7 2 (x) = x, i.e. 7 is a reflection. Let x C y which means ay = ax + 2~lu with suitable u £ P and 0 < a < 2"1. This yields ay + (I — o)0 = ax + -u + ( -- o)0 £

£

and 07(7;) + (1 — 0)7(0) = 07(2) + ^7(u) + (\ — 0)7(0), which implies l + 2o

2o

2o, 2o

Consequently

l-2o

1 . .

1

...

1 -2o

262

P UMPL UN AND R OHRL

and

0:7(2;) = aj(y) + -u follows because P is preseparated. This shows that 7 is antitone. Obviously if a fixed point exists it is 2-17(0). As P is preseparated

implies 7(2~17(0)) = 2~17(0), i.e. 2-17(0) is a fixed point. The following result is the converse of 3.2, it characterizes those PCmodules which possess a convex effect algebra structure.

3.4. Proposition: Let P be a preseparated proper PC-module with a mapping 7 : P —> P such that

is satisfied, for all x € P, and let 2~1P be positive-order-convex with respect to " C ". Put 1 := 7(0) and define x®y, for x, y 6 P, by 1.

.

1

1

if 2~lx + 2~ly 6 2~1P. Then (P, ©, 0,1) is a convex effect algebra. Proof: One verifies the properties in 1.1 If x 0 y is defined, x,y € P, then

-(z®3/) = ^(y®x] and hence £ © y = y 0 x, i.e. (El). If (x ®y] ® z, x,y,z £ P, is defined, 3.3 implies -

- -

-0

1 ,

1,1 ,

-

- --(

hence 1,1

1

1,1,

.

I s

ALGEBRAS AND MODULES

263

and \y+\z\2 \((x ®y)®z) follows because P is order preseparated (3.3, (iii)). This implies that y ® z is defined because 2~1P is positive-orderconvex. By an analogous reasoning it is shown that x 0 (y ® z) is defined and the equation (x ® y) ® z = x @ (y ® z) is trivial i.e. (E2) is proved. (E3) follows from the equation of 7 if one puts x' := i(x). To show (E4) let x 0 1 be defines, x £ P. Then

and

follow, hence

As P is preseparated this implies x = 0 i.e. (E4). (Cl) - (C4) follow directly from the Definition 2.1 of a PC-module. The order relation "c" on a PC-module will be called "canonical" . The full subcategory of PC generated by the PC-modules partially ordered by "C" will be denoted by Po^ -PC and may be also regarded as a full subcategory of Po -PC. Let Pi,i = 1,2, be PC-modules satisfying the hypotheses of 3.4. A positively affine mapping / : PI —» PI is called a morphism if /(7i(0)) = 72(0) holds, 7, : Pj —> Pj,i = 1,2, the mappings in 3.4. This is equivalent to the equation f ( - j i ( x i ) ) = 72(7(^1)), for any x\ € PI. The PC-modules satisfying the assumptions of 3.4 and these morphisms form the category

POC-PC;S. For any P e Poc-PC*S one defines A(P] := (P,©,1,0) (cf. 3.4), and any morphism / : P! —> P2 in Poc -PC*S is also a morphism / : A(Pi) —> A(P'2) of convex effect algebras and will be denoted by A(f). A : Poc-PC*s -> Conv-EffAlg is a functor. As 3.2 shows the functor M : Conv-EfFAlg —> Po-PC may be regarded as a functor M : Conv-EffAlg -> Poc -PC*S. Combining 3.2 and 3.4 yields the 3.5. Theorem: A : Poc -PC*., -> Conv-EfFAlg and M : Conv-EfFAlg -> Poc -PCps are inverse to each other.

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FUMPLUN AND ROHRL

Hence, investigating the theory of convex effect algebras, i.e. ConvEffAlg, is the same as investigating a category of preseparated proper positively convex modules supplied with a special type of affine automorphism of order 2. It could be interesting to examine which results on positively convex modules (cp. e.g. [5], [7], [8], [10], [11], [12], [13]) carry over to this special type of positively convex modules with the enriched structure of this special kind of automorphism. References [1] Flood, J., Semiconvex Geometry, J. Austral. Math. Soc. 30, 496-510 (1981). [2] Gudder, S., Convex structures and effect algebras, Int. J. Theor. Phys. 38, 12, 3179-3187 (1999). [3] Gudder, S., and Pulmannova, S., Representation theorem for convex effect algebras, Comm. Math. Univ. Carolinae 39, 4, 645-659 (1998). [4] Gudder, S., Pulmannova, S., Bugajski, S., Beltrametti, E., Convex and linear effect algebras, 44, 3, 359-379 (1999). [5] Kemper, R., Positively convex spaces, Appl. Cat. Str. 6, 333-344 (1998). [6] Mayer, P., Positiv konvexe Raume, Ph. D. thesis, FB Mathematik, FernUniversitat, Hagen (1993). [7] Nortemann, S., The Hahn-Banach Theorem for partially ordered totally convex, positively convex and superconvex modules, Appl. Cat. Str. 10, 417429 (2002). [8] Pumpliin, D., Regularly ordered Banach spaces and positively convex spaces, Res. Math. 7, 85-112 (1984). [9] Pumpliin, D., and Rohrl, H., Banach spaces and totally convex spaces I, Comm. in Alg. 12, 953-1019 (1985). [10] Pumpltin, D., and Rohrl, H., Convexity theories IV, Klein-Hilbert parts in convex modules, Appl. Cat. Str. 3, 173-200 (1995). [11] Pumpliin, D., Positively convex modules and ordered normed linear spaces, J. Conv. Anal., 10, 1, 109-128 (2003). [12] Wickenhauser, A., Positively convex spaces I, Seminarberichte, FB Mathematik, Bd. 30, 119-172, Hagen (1988). [13] Wickenhauser, A., Positively convex spaces II, Seminarberichte, FB Matheraatik, Bd. 32, 53-104, Hagen (1988). [14] Wong, Y.-C., and Ng, K.-F., Partially ordered topological vector spaces, Clarendon, Oxford (1973).

FIBREWISE SOBRIETY

GUNTHER RICHTER1 AND ALEXANDER VAUTH2 Fakultat fur Mathematik Universitat Bielefeld, Universitdtsstrasse 25, D-33615 Bielefeld, Germany E-mail : guenther.jutta,[email protected] E-mail : alexander. vauth@uni-bielefeld. de Fibrewise topology deals with so called total spaces X over a fixed base space T with respect to a continuous map p : X —> T, the projection. When T is a onepoint space the theory specializes to that of ordinary topology. There are fibrewise versions of many familiar topological concepts like separation, compactness, local compactness, even quasi local compactness, i.e. every neighbourhood U of a point contains a neighbourhood V that is relatively compact in U. This paper offers a notion of fibrewise sobriety and a fibrewise version of the Hofmann-LawsonTheorem, stating that every quasi locally compact sober space is locally compact. Moreover, Niefield's result on exponentiability of projections in the category of all sober spaces over a fixed sober space T generalizes to fibrewise sobriety. Mathematics Subject Classifications 54C10, 54D45

(2000): 18A20, 18B30, 54B30,

Keywords: fibrewise topology, fibrewise sobriety, local quasi-paracompactness, fibrewise Hofmann-Lawson-Theorem, exponentiable fibrewise sober maps. 0. Introduction

Fibrewise topology deals with so called total spaces X over a fixed base space T with respect to a continuous map p : X —> T, the so called projection. These are just the objects of the slice-category Top/y. When T = 1, the singleton, the theory reduces to that of ordinary topology. The fibres of p are the preimages Xt := p-1(i) of points t & T. More generally, Rw ••= R D p~l(W) denotes the fibre of R over W for R C X, W C T and Rt := R{t}There is a famous book by I.M. James on this subject [14]. It contains numerous fibrewise counterparts of ordinary topological notions and results. Most of them are simply obtained by passing from filters T to to called tied filters (f, t), i.e. filters f on the total space X, the p-image p(f) of which converges to t € T. For instance: 265

RICHTER AND VAUTH

266

Top 9 X X is a To-space

Top/T 9 p • X -> T p is fibrewise

(x € lim.F <=>• x' G lim^7) for all filters f on X => x = x' (lim J7 denotes the set of all limit points of J-)

(x 6 (lim^t <=> (lim.F)t) for all tied

is a Hausdorff space

| lim T\ < I for all filters f on X

some U &K(x),U'

&U(x')

x' e niters

The fibres Xt of p are T0-spaces p is fibrewise Hausdorff or separated | (lim ^t | < 1 for all tied filters OF, t) on X

x ^ x' in Xt =* U n C/' = 0 for some 17 e U(x), U' £ U(x') The diagonal AX is closed in the fibre product X XT X

The diagonal AX is closed in the product X x X (U(x) denotes the neighbourhood filter of x) X is compact

p is fibrewise compact

Every filter T on X has an adherence point

Every tied filter (jF, i) on X has an adherence point in Xt

Every open cover of X contains a finite subcover

Every open cover of a fibre Xt in X contains a finite subcover of some Xw with W € U(t) p is proper, i.e. closed with compact fibres [5] p is (pullback-)stably closed [8]

FIBREWISE SOBRIETY

X is locally compact Every neighbourhood [7 of a point x £ X contains a neighbourhood V that is compact, i.e. every filter T on V has an adherence point in V

X is quasi locally compact or core-compact [12]

267

Top/T 9 p : p is fibrewise locally compact [20] Every neighbourhood U of a point a; € X contains a neighbourhood V such that for some W € U(p(x)} every tied filter (.T7,*) on V with t e W has an adherence point in Vt Every neighbourhood U of a, point x 6 X contains a neighbourhood V such that for some W £ W(p(o;)) with p(V) C W the (co-)restriction p\ff : V —> W is proper p is fibrewise quasi locally compact or fibrewise core-compact [18]

Every neighbourhood 17 of a point x & X contains a neighbourhood V that is relatively compact in [7, i.e. every filter on V has an adherence point in U

Every neighbourhood U of a point x e X contains a neighbourhood V such that for some W € U(p(x]) every tied filter (JF,t) on V with t e W has an adherence point in C/t

X is exponentiable in Top, i.e. — x X is left adjoint

p is exponentiable in Top/r, i.e.,— x p is left adjoint (p' X p is the canonical map

All fibrewise notions mentioned above are pullback-stable, compositive and, of course, closed under composition with isomorphisms. For fibrewise local compactness this follows from Theorem 0.1. LetP be a pullback-stable and compositive class of continuous maps which is closed under composition with isomorphisms. Say that a continuous map p : X —> T has the local P-property if every neigh-

268

RICHTER AND VAUTH

bourhood U of a point x e X contains a neighbourhood V such that for some W 6 U(p(x}) withp(V) C W the (co-)restriction p\$ : V -> W ofp belongs to P. Then the class of all p with the local P-property is pullback-stable, compositive, and closed under composition with isomorphisms as well. Proof. Consider the following pullback in Top

S xTX

X

/ with 5 x r X = {(s,x) e S x X \ f ( s ) = p(x)}, q the first and g the second projection. Assume the local P-property for p. In order to prove it for q, take ( s , x ) € S XT X and a basic neighbourhood O XT U 6 U(s,x), O € U ( s ) , U 6 W(x). By assumption on p, there are neighbourhoods J7 D V 6 W(x) and W € U(p(x)) such that p(K) C H^ and p|^ e P. Now look at the following commutative cube, the front, top, bottom, and hence the back of which are pullbacks:

OxTX

By assumption on P, the (co-)restriction q' = q\Q^Ty belongs to P. Moreover, O X T U I> O X T V e W(s,x) and O n /"H^O e W(s), because f(s)=p(x). Next consider a composition Y —> X —> T of continuous maps and assume the local P-property for both, p and g. In order to prove it for p o
FIBREWISE SOBRIETY

269

By assumption on p, there are U 15 V £ U(q(y)) and W 6 U(p(q(y}}~) such that ply' e P. Now consider the pullback of g|p along the embedding V <-» C/ and compose it with ply':

D

Another example for P is the class of all separated projections. The corresponding local P-property for p : X —> T is to be locally separated, which simply means that every point in X has a neighbourhood U such that p u is separated [8]. The first section introduces a notion for fibrewise sobriety that is both, pullback-stable and compositive. It generalizes H. Herrlich's categorical characterization of the category Sob of all sober spaces as reflective hull of the Sierpinski space 2 formed in the category Top0 of all To-spaces [9]. The second section deals with a generalization of the Hofmann-LawsonTheorem, stating that quasi locally compact sober spaces are locally compact [12]. The last section 3 extends Niefield's characterization of the exponentiable objects in the slice-category Sob/V of all sober spaces over a fixed sober space T [7, 16, 17] to the category (Top/x)s of all fibrewise sober projections with an arbitrary base space T. Categorical terminology follows [1]. 1. Fibrewise Sobriety

Recall that a subset A of a space X is irreducible iff O n A ^ 0 / O' fl A implies (O n O') D A ^ 0 for all open subsets O,O' C X. Any set A between a singleton {#} and its closure {x}~ is irreducible. A space is called quasisober (sober) if every nonempty closed irreducible subset has a (unique) generic, i.e. dense point. A sober space is automatically T0. Quasisober T0-spaces are sober. The full subcategory Sob of all sober spaces is wellknown to be reflective in Top, the category of all topological spaces [2]. The most familiar description of sober spaces in terms of filters uses completely prime open filters, i.e. proper filters U of open sets such that UV G U implies V 6 U for some V € V where V is a set of opens. Examples

270

RICHTER AND VAUTH o

are the open filters U(x) of open neighbourhoods of a point x. Quasisoberness (Soberness) means, that these are the only ones (and U(x) = U(x') implies x = x'}. Passing from filters to tied filters suggests the following fibrewise version of soberness for a projection p : X —> T: (U, t) completely prime open tied filter on X =>• U = U(x) for exactly one x € XtPhrased in terms of irreducible sets, this means: X ^ A = A~ irreducible and t € p(A)~~ =>• A = {x}~ for exactly one x <=Xt. Fortunately, this property implies soberness of the fibres and turns out to be compositive. Unfortunately, it is not pullback-stable, in general, because the identity p = id T : T —> T appears as pullback of 1 —> 1 along T —> 1 and shares the property iff T is symmetric, i.e. t' € {t}~ <*=> t e {t'}~, and quasisober. The picture changes in case of projections with sober fibres. They are pullback-stable and, moreover, form a reflective subcategory of Top/ T [20]. But they fail badly to be compositive, as the composition of the embedding of a non sober subspace X in a sober space T with T —» 1 shows. There is a better possibility inbetween. Definition 1.1. Call a projection p : X —» T fibrewise sober iff for any closed irreducible subset A C X and t £ T with p(A)~ = {t}~ there exists exactly one x € Xt such that A = {x}~. Theorem 1.2. The class of all fibrewise sober projections is pullbackstable, compositive, and closed under composition with isomorphisms, Proof. Consider the pullback (*) as in the proof of Theorem 0.1, an irreducible closed subset C C 5 XT X with q(C)~ = {s}~ for some s e S, and let p be fibrewise sober. Then g(C)~ is an irreducible closed subset of X and p(g(Crr = p(g(C)}- = /(<7(C))~ = f(q(C)-)- = /({*}-)- = { f ( s ) } ~ - By assumption on p, there is a unique x £ ^f(s) such that Note that x e ^/(s) implies (s,z) £ S XT X. Moreover, ( s , x ) e C, o

o

because for any basic neighbourhood O XT U of (s, x), O 6 U(s), U £ U(x)

FIBREWISE SOBRIETY

271

one gets O n q(C) ^ 0 ^ U n s(C) by X -^> T of fibrewise sober projections and an irreducible closed subset C C Y with p(q(C))~ — {t}~ for some £ € T. Then q(C)~ is an irreducible closed subset of X wi£hp(q(C)~)~ = {t}~. By assumption on p one gets a unique x € Xt such that q(C)~ = {x}~ as well as a unique T/ € Yx with (7 = {y}~, by assumption on q. If C = {y'}~ with p(q(y')) = t then g(C)~ = {q(y')}~, hence g(y') = x, by uniqueness of x and y = y' by uniqueness of y. D Corollary 1.3. The fibrewise Sierpinski space over T, namely the second projection ^2 : 2 x T —> T, is fibrewise sober. Proof. One gets 7r2 by pulling back 2 —» 1 along T —> 1.

D

Corollary 1.4. (Top/T)s = Sob/T iff T is a sober space. Proof. Note Sob/T C (Top/T)s for T e Sob, consider the composition X -^ T -> 1, and observe id T e (Top/T)s. D Theorem 1.5. (Top/T)s is reflective in Top/T. Proof. For Top/T 9 p : X —> T and X 2U open consider C> := {(A, £)|X D A = A- irreducible, p(A)~ = {t}", and [7n A ^ 0}. The assignment [7 H-» [7 is easily seen to preserve arbitrary unions and finite intersections. Therefore, {U\X 2 U open} is a topology on X. Moreover, the second projection p : X -» T fulfills p~1(W) = Xw for T D W open, which implies continuity. The assignment X 9 x H-> ({x}~,p(o:)) e X defines a map aX : X -> X with p o crX = p which is continuous, because (
272

RICHTER AND VAUTH

In fact, (aX)~l yields an isomorphism of topologies. Consequently, (crX)"1 preserves irreducible closed subsets X \ U C X, especially those with p(X x 17) ~ == {t}~ for some t £ T. For any open subset W C T one gets

Z7) n WV 0 «=^ (x \ [/) n xw ^ 0 U <=» (X x 10 n

n w ^ 0. This shows p(X \ U)~ = p(X \ Z7)~ = {t}~, hence (X \ U,t) e X. For any open subset O C X observe

u, t}} n 6 ^ 0 <=» (X \ u, t) e 6 <=» (X \ [/) n o ^ 0 This shows X ^ U = {(X \ C/,t)}-, and for X \ L7 = {(X \ U',t)}~ one gets (X \ U) n O ^ 0 4=> (X \ C/') n O ^ 0, hence X\U = X\U'. Altogether, p turns out to be fibrewise sober. Now take q : Y —> T in (Top/x)s and a continuous map / : X —> V with ^ o / = p. For (^,i) e A" observe g(f(A)-)~ =p(A}~ = {t}~, f(A)~ being irreducible. This yields a unique y £ Yt with f(A)~ = {y}~. Now define f ( A , t ) := y. Then q(f(A,t)) = t = p(A,t) as well as f ( a X ( x ) ) = f({x}-,p(x)) = f ( x ) , because f({x}-)- = {f(x)}~ and /(x) € Yp(x). A

A

O

A

In order to prove continuity of / consider (A, t) e X and V € U(f(A, t)). Then t/ := / -1 (y) is open in X and 7 n f ( A ) ^ 0 by /(A,i) e f ( A ) ~ , hence C/n^ ^ 0, thus (A, t) e C7. For any (A1, t') 6 U observe f ( A ' , t') e Yt> with {/(^'.t7)}- = /(>!')". P(AT = {*'}-, and C/ n A' ^ 0. This implies 0 ^ /([/ n .4') C /([/) n /(^') C y n /(A') and for some z e V n /(A') the following holds

This shows /(C/) C V, hence continuity of / at (A, i). Uniqueness of / follows from aX(A)~ = {(A,t)}~ for (A,t) & X, because for any open U C X one gets aX(A] n U ^ 0 «=> o-X(a) = ({a}~,p(a)) e U for some a e A a e U for some a e A «=> (7 n A ^ 0

FIBRE WISE SOBRIETY

273

For any continuous maps / : X —> Y with / oaX = f and q o f = p observe f ( A ) ~ = J(aX(A))- = ](aX(A)-r = J({(A,t)}~r = { f ( A , t ) } ~ , hence f ( A , t ) = f ( A , t ) by uniqueness of the generic point of f ( A ) ~ in Y^. D

Remark 1.6. As already mentioned in the proof above, (aX}~1 induces an isomorphism between the topologies of X and X. Consequently, if / : X —> Y is initial, the same holds for /, because for any open U C X one gets an open subset V C Y such that (aX)~l(U) = U = f~l(V) = ((TX)-^/- 1 ^)). This implies U = f~l(V). Moreover, if / is an embedding or simply an inclusion of a subspace X t—*Y, then / is an embedding as well, because f(A, t) = f(A', « ' ) = » * = P(A, t) = q(f(A, t)) = ...=t' and

The b-topology on a space X e Top is generated by its locally closed subsets O f~l C, O open, C closed. A continuous map / : X —» Y is called b-dense, if it is dense with respect to the fr-topology on Y [4]. Lemma 1.7. The reflection morphisms aX : X —> X are b-dense.

Proof. Consider some locally closed subset Ur\(X\V) = U\V^ = 0. For x 6 t/ \ V observe {x}~ n C/ ^ 0 and {x}~ n V = 0, hence aX(:r) = ({o;}-,p(a ; ))6f7\f. D Lemma 1.8. Restrictions of fibrewise sober projections q : Y -* T to b-closed subspaces X C Y are fibrewise sober. Proof. Consider an irreducible closed subset A C X such that q(A)~ = {t}~. Then A~ = {y}~ in F for a unique y G yt. Now y & X, because for any V £ U(y) one gets

({y}- r\V)r\x = A- r\vr\x = Ar\v^<&, according to y € A~. Thus A = [y}~ n X with y E Xt. Uniqueness of y is obvious. D

274

RICHTER AND VAUTH

Proposition 1.9. Let p : X -» T be the restriction of a fibrewise T0projection q : Y —> T to a subspace X C Y. Then X is b-dosed if p is fibrewise sober. Proof. Consider y<=Y such that ({y}~ n V) n X ^ 0 for any V 6 and show y e X. Now A := {y}~ n X j= 0 is closed in X and even irreducible, because for open sets O, O' C V one gets

o n A ^ 0 ^ O ' n A = » i / e O n O ' e W(y) => (O n O') n 4 ^ 0. Moreover, p(A)~ C T in Top/T to b-closed subspaces. Proof. Just as 2 in Top0, the fibrewise Sierpinski space 7r2 turns out to be a coseparator in (Top/r) 0 . By Corollary 1.3, Theorem 1.5, and Lemma 1.8 all restrictions of powers of 7r2 to 6-closed subsets are fibrewise sober. Vice versa, any fibrewise sober projection is fibrewise TO, hence it admits an embedding into a power of 7r2 , the image of which is 6-closed by Proposition 1.9. D Just as in [4] for T = 1, the epimorphisms in (Top/V)o turn out to be the b-dense ones. Lemma 1.11. The following are equivalent for (Top/r)o B p : X —> T, q : Y —> T and a continuous map f : X —> Y: (i) f is b-dense. (ii) f is an epimorphism in (Top/T)oProof, (i) =>• (ii): Consider continuous maps g, h : Y —* Z, r : Z —> T in (Top/r)o with g o f = h o / and rog = q = roh. Assume g(y) ^ h(y) for some y £ Y. By g(y), h(y) € Zq(y} & Top0 one gets g(y) <£ {h(y)}~

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or h(y) i {g(y)}~. If the latter holds, then h~l(Z \ {g(y)}~) € and one gets some x e X with /(a;) € {y}~~ fl h~l(Z \ {p(j/)}~). Now ) = 0(/(z)) e g({y}~) !~\(Z\ {g(y)}~) = 0, a contradiction. (ii) => (i): Let C C Y be the 6-closure of f(X) and consider the pushout

over T. By its universal property one gets a continuous map k : Z —> Y with fcog = idy = fco/i. Using this, r turns out to be fibrewise TO, because z,z' € Zt with k(z) ^ k(z') can be easily separated. For k(z) = k(z') =: y and z ^ z' observe y $. C using k~l(y) = {g(y},h(y)} according to Z = g(Y) U h(Y) and the definition of k. For y £ C there is y € U(y) such that {y}~ n y n C = 0, because C is 6-closed. N o w y n C c y \ { j / } - and g(yuy\{7/}-)U/i(y\ {?/}-) =: W turns out to be open in Z which carries the final topology with respect to g and h. Moreover g(y) € W and h(y) $ W. Finally g o / = h o / implies g = h, hence C = Y. D Using Theorem 1.10 and Lemma 1.11, (Top/T)s appears as the (epi-)reflective hull of 7r2 in (Top/T)o- In case of T = 1 one gets Sob as (epi-)reflective hull of 2 in Top0, as it was observed by H. Herrlich [9]. Moreover, a characterization of sober spaces by R.-E. Hoffmann [11] generalizes as follows: Corollary 1.12. A projection p : X —> T is fibrewise sober iff every (Top/T^o-epimorphic embedding f : X —> Y is an isomorphism. The three possible fibrewise versions of sobriety discussed in the beginning of this section turn out to be closely related. Recall that T is a TD -space if its points are locally closed [3, 6]. Theorem 1.13. Lei T be a fixed base space. For any projection p : X —> T consider the following properties:

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(1) X D A = A~ irreducible and t 6 p(A)~ =>• A = {x}~ for exactly one x € -X"*. (2) p is fibrewise sober. (3) The fibres of p are sober spaces. Then (1) =>• (2) =^ (3). Moreover, (3) => (2) /or every p •<=>• T is a TD -space, (2) => (1) /or every p <£=>• T is symmetric and quasisober, (3) =>• (1) /or every p <*=> T zs T\ and sober. Proof. (1) =>• (2) =» (3) is obvious. If (3) =$• (2) holds for every p, then (2) holds for the embedding p : T \ {t} <-> T. Now {£} is locally closed iff A := {t}~ \ {t} is closed. If not, one gets A~ = {t}~ and every neighbourhood of t meets A. Therefore, A turns out to be irreducible and closed mT ^ {t} with p(A)~ = {t}~. But the fibre over t is empty, a contradiction! Conversely, let T be a T^-space, p : X —> T with sober fibres and A an irreducible closed subset of X with p(A)~ = {t}~ for some t € T. By assumption on T, there is an open W C T such that {i}~ f~l VF = {i} and {t}~ \ {t} is closed. This yields t e p(A) and At = XV n A jf= 0. Moreover, At is irreducible, because O n At ^ 0 ^ O' n At for open subsets O,O' C X implies (O n Xw) n A ^ 0 + (O1 r\ Xw) r\ A, hence 0 ^ (O n O' n Xw) C\ A = (O Ct O') C\ At. By assumption on p, there is a unique x € Xt such that {x}~ n Xt = At, hence At C {x}~ C A. Furthermore, ((A" \ (o;}~) n XH/) n A = 0 and Xw n A = At ^ 0 implies (X \ {x}~) n A = 0, thus A C {i}-. If (2) =»• (1) holds for every p, then (1) holds for id T : T -> T. This forces T to be symmetric and quasisober. Vice versa, p(A)~ — {t}~ for all t € p(A)~ shows (1) for p with (2). What remains is immediate, because ((3) =* (1)) iff (((3) =» (2)) and ((2) => (1)))

according to (1) => (2) =>• (3) and T is TI iff it is TD and symmetric.

D

2. The Fibrewise Hofmann-Lawson-Theorem This section provides a fairly general but sufficient condition on the base space T which enables to show that fibrewise quasi locally compact and

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sober projections p : X —> T are fibrewise locally compact. The difficulty is, to get certain (co-)restrictions of p closed. This problem does not arise in case of T = 1 . Definition 2.1. For R, S C X, W C T say that R is fibrewise way below S over W, in symbols R <^w S, iff every tied filter (F, t) on R with t £ W has an adherence point in the fibre St. Recall from [18], that R
R <w V <w U. Unfortunately, W may be strictly smaller than W, in general. Definition 2.3. Say that p : X —> T has a stationary fibrewise interpolation-property iff every t £ T has a neighbourhood base of sets C such that R T be fibrewise quasi locally compact with quasisober fibres and stationary fibrewise interpolation property. Then p is fibrewise locally compact. Proof. For x e X, U £ U(x) there are V £ U(x), W e U(p(x)) such that V -€.w U. Moreover, there is W 2 C £ U(p(x)} and a descending chain of open subsets Vn C Xc with

Vn+l Now JiT := (n^n)c has compact fibres Kt, t € C, because Kt appears as N

intersection of the Scott-open filter Ut = IJ{^l(^n)t £ O open in Xt} in

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VAUTH

the quasisober fibre Xt. Moreover, Ut is the open neighbourhood filter of Kt in Xt [13, 15]. The (co-)restriction p\% of p turns out to be closed, because for K I> A closed in K there exists an open set O C X with O n K = K \ A. Now #4 C O for £ e C\p(4), hence (Vn)t C O for some n € N, thus 04+i)vK' C O for some neighbourhood W' of t in C, which implies KW C (9 and W C C \ p(A) e Z/(t) in C. D Proposition 2.5. Tfte following are equivalent for a space T: (i) Each open cover has a locally finite shrinking. (ii) T is a T^-space and each open cover has an open locally finite refinement. (Hi) For every open cover (C/i)/ there is an open locally finite cover (Wj)j such that (W~)j refines (C/i)/. Proof, (i) => (ii) : T is a T4-space iff each open locally finite cover is shrinkable [10]. A locally finite shrinking of an open cover is both, locally finite and an open refinement. (ii) =>• (iii): (C/i)/ has an open locally finite refinement (Vj)j which has a (again locally finite) shrinking (Wj)j. Now (W~)j refines (C/i)/. (iii) =>• (i): Let (C/i)/ and (Wj)j be as in (iii). Choose for each j & J some i(j) e / such that Wj~ C U^j) and define Vi := U{W^-|i(.7') = i}, i e /. Then (Vi)/ is an open locally finite cover. Moreover, V~~ = \j{W~\i(j) = i} C C/i, because (Wj)j is locally finite. D Recall for instance from [10] that T is paracompact iff it is Ti and fulfils 0).

Definition 2.6. Call a space T quasi-paracompact iff it fulfils (one of) the conditions in Proposition 2.5 and locally quasi-paracompact, if each point has a neighbourhood base of quasi-paracompact subspaces. Theorem 2.7. Let T be locally quasi-paracompact and p : X —» T fibre-wise quasi locally compact with quasisober fibres. Then p is fibrewise locally compact. Proof. Show that p has a stationary fibrewise interpolation-property and apply Theorem 2.4. To this end consider some quasi-paracompact neighbourhood C C T and open subsets R, U C Xc such that R T, the (co-)restriction p|£c

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of p inherits fibrewise quasi local compactness. Therefore, for each t e T, interpolation yields open subsets V* C Xc, W* C C with t e W* such that D

„

T/t

-^

TT

fi ^vi" " <4~.IV* U •

By assumption on C, the open cover (W*)tec has a locally finite shrinking (O*)tec in C1- Now choose V := (j VQI and prove R -Cc V ^c U. For c the first part R
/ iv

= UVo*rw = \JV&tnW C (Jl/*. Moreover, t e S
£•

E

implies to € O*~ n C C W*. Now consider an arbitrary open cover W of t/to in Xc- By V* Ciy* ^ for all t € E, there exist an open neighbourhood VF of to in C and a finite subset £ C U such that U£ D ( U V* I . The latter contains V \E

~ and

) jy

W n W is a neighbourhood of to.

D

3. Exponentiable Fibrewise Sober Maps For sober Jb-spaces T and Sob/r 3 p : X -> T Niefield [16, 17] has shown, that p is exponentiable in Sob/r iff it is exponentiable in Top/T- This is by no means obvious, because - x p to be left adjoint implies preservation of coequalizers. In general, the latter are neither final nor surjective in Sob/T • The aim of this section is to establish Niefield's result for arbitrary base spaces T and (Top/T)s instead of Sob/T- Recall from Corollary 1.4 that (Top/T)s = Sob/T for a sober base space T. Just as for Top0 in Top there is a reflector R : Top/T —> (Top/r)o given by identification of points with equal closure in the same fibre. The corresponding reflection-morphisms are open and establish an isomorphism between the topologies of the total spaces. If 5 : Top/T —* (Top/T)s denotes the reflector constructed in the proof of Theorem 1.5, then S = SoR. Furthermore, the reflection-morphisms crX : X —> X are embeddings and (Top/T)o-epimorphisms for fibrewise Tb-projections p : X —> T.

280

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Lemma 3.1. R preserves finite products. Proposition 3.2. S(qxp] — S(q)xp forp in Top/T, and arbitrary q & Top/T •

(Top/T)s, p exponentiable

Proof. S(qxp) = S(R(qxp)) = S(R(q)xp) and S(q)xp = S(R(q))xp. Therefore, it suffices to consider the case q 6 (Top/T)o- Now look at the following pullbacks over T, Y xTX

YxTX

Y

•Y where a denotes the reflection-morphism for q x p, the universal property of which yields T with T o a = aY XT X, because q x p : Y XT X —> T is fibrewise sober. By assumption on q, crY is an embedding. Therefore, aY XT X, hence r share this property by Remark 1.6. Moreover, aY is an (Top/T)o-epimorphism and p is exponentiable in (Top/T)o as well [19]. Consequently, aY x p — aY XT X is an (Top/r)o-epimorphism as well as T, which proves r to be an isomorphism by Corollary 1.12. D Theorem 3.3. Let p £ (Top/V)s be exponentiable in Top/T, then p is exponentiable in (Top/V)s. Proof. Obviously, coproducts of fibrewise sober projections are again fibrewise sober. Consequently, — x p : (Top/T)s —> (Top/T)s preserves coproducts. Moreover, it preserves coequalizers, as the following pullbacks show, where c is the coequalizer of (Top/r)s-morphisms /, g in Top/T, hence aC o c their coequalizer in (Top/T)5:

CxTX

/

•c

By assumption onp,cxx^ = cxpisthe coequalizer of / XTX = f xp and = gxp'm Top/T, hence a o (c x T X) is the respective coequalizer in

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281

(Top/r)s. By Proposition 3.2, the same holds for (aC X T X ) o (c X T X ) — (aC o c) XT X = (aC o c) x p. Now the Special Adjoint Functor Theorem applies, because (Top/r)s has a separating set ({f £ T\{t'}~ — {t}~} <—> T)teT and is cowellpowered. The latter follows from cowellpoweredness of (Top/T)o, because (Top/r)s is mono-reflective in (Top/T)o- Therefore, the embedding (Top/r)s •—> (Top/V)o preserves epimorphisms. In order to prove that (Top/7-)o is cowellpowered, use Lemma 1.11 and the 6-topology-functor from (Top/r)o to the full subcategory Sep of all separated p £ Top/y, which is faithful and preserves epimorphisms. D The elementary proof for nbrewise quasi local compactness of an ex9 f ponentiable p € Top/^ in [19] uses a certain test-quotient P-»Q—>T, Q and P being 0-dimensional To-spaces, hence Hausdorff, especially sober. Nevertheless, P and Q may fail to be nbrewise sober over T. This depends heavily on T, because for each t £ {f(x)}~ \ {/(z)} wi*h {*}~ — {/(x)} one has to add a point ({a;}, t) to Q in order to get Q. Fortunately, S(q) : P —> Q is again quotient, because S(q)({x}, t) = ( { q ( x ) } , t ) proves S(q) to be surjective. Furthermore, for any V £ Q with S(q)~1(V) = U, U open in P, U turns out to be saturated with respect to q, i.e. U = q~l(q(U)). Hence V := q(U) is open in Q and one easily obtains V = V which proves S(q) to be final. ~

A

Now take S(q) instead of q and consider the following pullbacks in (ToP/r)s PxTX

S(q)xTX

„

5

*QxTX-^-*X

The map S(q) XT X is surjective in any case. If p is exponentiable in (Top/r)a, S(q) XT X = S(q) x p must be the coequalizer of the (— x p)image of a pair of (Top/T)s-rnorphisms in (Top/r)«. Therefore, if / x p : QXTX —> T remains to be fibrewise sober with respect to the final topology induced by S(q) x p, it carries already this final structure. To check this, consider Q XT X 3 A = A~ irreducible in the final structure with (/ x p)(A)~ = (f o s)(A)~ = f(s(A)-)~ = {t}~. Then

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282

there is a unique y € Q with {({y},t)}~ = s(A)~ in Q. This implies A C {({y},t)}~ XT X, which is closed in the pullback structure. Moreover, {({y},t)}~ = {({y},t')\{t'}~ = {t}~} is an indiscrete subspace of Q that is isomorphic to {({z},t)}~ in P for some z G P with
{({y},t)}~xTx

{({z},t)}-xTX

QxTX

PxTX

{({y},*)}S(g)

•Q

This shows {({z},t)}~ XT X = {({y},t)}~ XT X for both structures, thus {({y})*)}" XT X remains unchanged and A turns out to be closed (and irreducible) in the pullback-structure. Consequently, there exists a unique ),z) e (Q xTX)t with {(({y},t),x)}- = A. Finally observe, that the first projection TTQ : Q —> Q, ({y},t) H-> y, is surjective, continuous, and open as well as the first projection np : P —> P, Then TTQ XT X and TTR XT X have the same properties and (TTQ XT X) o (S(q) XT X) = (TTQ o S(q)) XT X = (q o irP) XT X = (q XT X) o (KP XT X) proves q XT X = q x p to be a quotient. This suffices to show that p is fibrewise quasi locally compact, hence exponentiable in Top/r [19]. Theorem 3.4. Letp be exponentiable in (Top/V),s, thenp is exponentiable in Top/TRemark 3.5. By the same methods, the exponentiable objects in the full subcategory of all projections p £ Top/T with sober fibres turn out to be exponentiable in Top/T and vice versa. It is not necessary to pass from q to S(q) in this case. References [1] J. Adamek, H. Herrlich, G.E. Strecker: Abstract and concrete categories, John Wiley (1990).

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[2] M. Artin, A. Grothendieck, J.L. Verdier: Theorie des topos et cohomologie etale des schemas, Lecture Notes in Math. 269, Springer (1972). [3] C.E. Aull, W.J. Thron: Separation axioms between TO and TI, Indag. Math. 24 (1963), 26-37. [4] S. Baron: Note on epi in TO, Canad, Math. Bull. 11 (1968), 503-504. [5] N. Bourbaki: Topologie generate, Hermann (1961). [6] G. Bruns: Darstellungen und Erweiterungen geordneter Mengen I., II., J. reine angew. Math. 209 (1962), 167-200, 210 (1962), 1-23. [7] F. Cagliari: Cartesian objects and exponentiable morphisms in topology, Rend. Circ. Mat. Palermo (2) Suppl. No 29 (1992), q25-40. [8] M.M. Clementine, D. Hofmann, W. Tholen: The convergence approach to exponentiable maps, Port. Math. 60 (2003), 139-160. [9] H. Herrlich: On the concept of reflections in general topology, in: J. Flachsmeyer, H. Poppe, F. Terpe (eds.), Contributions to extension theory of topological structures, Proc. of the Symp. held in Berlin 1967, VEB Deutscher Verlag der Wissenschaften, Berlin (1969), 105-114. [10] H. Herrlich: Topologie I: Topologische Raume, Heldermann (1986). [11] R.-E. Hoffmann: Charakterisierung niichterner Raume, Manuscripta Math. 15(1975), 185-191. [12] K.H. Hofmann, J.D. Lawson: The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285-310. [13] K.H. Hofmann, M.W. Mislove: Local compactness and continuous lattices, in: B. Banaschewski, R.-E. Hoffmann (eds.), Continuous lattices, Lecture Notes in Math. 871, Springer (1981), 209-248. [14] I.M. James: Fibrewise Topology, Cambridge tracts in mathematics, no. 91 (1989). [15] K. Keimel, J. Paseka: A direct proof of the Hofmann-Mislove theorem, Proc. Amer. Math. Soc. 120 (1994), 301-303. [16] S.B. Niefield: Cartesian spaces over T and locales over fi(T), Cahiers Topologie Geom. Differentielle 23 (1982), 257-267. [17] S.B. Niefield: Exponentiable Morphisms: Posets, Spaces, Locales, and Grothendieck Toposes, Theory Appl. Categ. 8 (2001), 16-32 (electronic). [18] G. Richter: Exponentiability for maps means fibrewise core-compactness, J. Pure Appl. Algebra 187 (2004), 295-303. [19] G. Richter: An elementary approach to exponentiable maps, submitted. [20] A. Vauth: Faserweise nuchterne exponentielle Abbildungen, Diploma Thesis, Universitat Bielefeld (2003).

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TWO APPLICATIONS OF ELEMENTARY SUBMODELS TO PARTITIONS OF TOPOLOGICAL SPACES

J. SCHRODER Universiteit van die Vrystaat Departement van Wiskunde Posbus 339 Bloemfontein 9300, Suid Afrika E-mail: [email protected] Web: wwwl6.brinkster.com/jodis STEVE WATSON York University Department of Mathematics 4700 Keele St. North York, Ontario M3J1P3, Canada E-mail: [email protected] Web: math.yorku.ca/watson/ We are looking at the sizes of partitions of topological spaces and show: No quasilindelof space can be partitioned into more than 2N° sets Pj <E P, where the character of Pi in X is countable and any two distinct Pi,Pj 6 P can be separated by open sets with disjoint closure. No H-closed space can be partitioned into more than 2N° sets PJ e P, where the character of Pi in X is countable and any two distinct Pi, Pj S P can be separated by disjoint open sets. Mathematics Subject Classifications (2000): primary 54A25, secondary 03E75 Keywords: elementary submodel, partition, cardinal invariant, quasilindelof, H-closed

Introduction: The main area of application of elementary submodels in topology is in the theory of cardinal invariants. Here we are mainly interested in sizes of partitions of topological spaces. A prominent example is the Theorem of Archangelski-Pol: Theorem 1. No compact T% space can be partitioned into more than 2 N ° closed GS sets. 285

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We will obtain this theorem as a corollary. We start by quoting some facts which are pivotal for the application of elementary submodels to topology in general and for this article specifically. A detailed account of the general background can be found in [4]. Lemma 1. Let (j>(x, v\, V2, • • • , vn) be a formula of set theory with free variable x and Vi e V (the set-theoretic universe) and let A be a set. Then there exist a set M such that A C M, \M\ = \A\-w and if there are mi,rri2,... ,mn e M such that 3x : 4>(x,mi,m2,...,mn), then there is such an x in M.. Proof: Start with MO = A. Select an x if <j)(x,mi,m2, • • • ,mn), where m i , m 2 , . . . ,m n e Mk and put it into Mk+i- Set M := \jMk- O Remark 1. We may find M for finitely many formulas simultaneously. M. contains a witness of truth (or falseness) of <j). If our x is unique, it must be in M, of course. We say that elements definable by elements of M are in M.. Example 1.

a) 4>(x,v) :="x £ v" Hence, if m G M. and 3x : x G m, then there is such an x in M. M. witnesses (reflects) non-emptyness. b) (x,vi,v2):="x£vi\V2,n Take a set X, X & M, A & M, A C X. If X n M C A, then X = A. Otherwise there would be an m e M with m e X \ A. c) The formulae used for M. are absolute, i.e. they are true in M. iff they are true in V. Before we use M., we must be sure which finite set of formulae we need. In practice, we look at the proof and collect all formulae along the way. d) There is a formula which defines u\ uniquely. So we can create a countable M with first uncountable cardinal u^4 and then wf1 = wi. But u>\ fl M is countable and never element of M [using b), u>i, u>i H .M € M and ui\ n M. C wi n M. imply wi = w± n M, a contradiction]. e) Let P be a partition of X, P € M. and (/)(x,vi,V2) := nv\ e a: £ v2" • I f P r \ M = £ < & , then P £ M. Because, if there is a P & P such that m € P e P, then there is such a P in M; but P is unique, since it is member of a partition.

APPLICATIONS OF ELEMENTARY SUBMODELS

287

Lemma 2. Let X be a set, \X <2K. There isM^X, \M\ = 2K, M is closed under K-sequences and M. reflects 3x : , Set M := (JMa. D Lemma 3. Let K,X £ M, K C M. reflects 3o; :
If \X\ < K then X C M and M

Proof: (/>i(x,vi,V2) = "x : v\ —> v^, onto", (foO^i,^,^,^) = = s) A (VB : ui -» t>2 onto) A (^4 £ i>i)". D Definition 1. Let (X, #) 6e a topological space. a) (X, X) is called quasilindelof (qL), if every open cover {Ui\i € /} of X has a countable subcollection {Uin\n e IN} such that \J{d(Uin)\n&]N} = X. b) Let P CX. Then U(P) := {O\P C O £ X}. X(P) denotes the minimal cardinal of an open basis ofU(P). Theorem 2. No qL space (X, X) can be partitioned into more than 2N° sets Pi e P, where U(Pi) < KQ and for any two distinct Pi,Pj £ P there is UPi 6 U(Pi), Up. € U(Pj) such that dUPi n dUPi = 0 Proof: LetX,X,U,P € M and \M\ < 2 K °. XCiM is 6l-closed with respect to P: Let P e P and dUn n X n M ^ 0 for all Un € W(P). Select points xn e dC7 n nXnA1. Then (x n ) nej fv e ^Vl and (x n )jv determines P uniquely, hence P 6 M. Assume \J(PnM) ^ X. Then there is P0 e P such that P0nXnM = 0. For every P&P with PnXn.M ^ 0 there is t/0(P) £ U(P) with dC/0(P) n Po = 0. {f/o(P)|P n X n M 7^ 0} is an open cover of X n M and has a countable subcollection {[/0(P;)|£ 6 IV} as stipulated in Definition la). Since P; e M and P e X we have {[/0(P;)|/ 6 W} e X and U{^o(P;)l^ € ^VJ £ A4. A contradiction appears because of Example lb).D Corollary 1. No Lindelof regular T\ space can be partitioned into more than 2 N ° closed subsets with a countable base. D Corollary 2. No compact T2 space can be partitioned into more than 2H° closed GS sets. D Remark 2. We will now weaken the separation axiom and strengthen the covering property.

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Definition 2. (X, X) is called H-closed, if every open cover of X has a finite, dense sub-collection. Theorem 3. No H-closed space (X, X) can be partitioned into more than 2N° sets Pi £ P, where x(P«) 5: NO and for any two distinct Pi,Pj 6 P there is UPi € W(P»), Up, € U(Pj) such that UPi l~l UPj = 0. Proof: To every x e X assign an open ultrafilter F(x) such that F(x) —> x. Define a set A C X to be s-closed with respect to P if VQ € P[(VC/g € U(Q)3a £ A : UQ £ F(a)) =» Q H A ^ 0]. X n Ails s-closed w.r.t. P: Let P e P and select for each Un £ W(P) an xn e X n A4 such that f/n 6 F(xn). Then (x n ) ne jv £ Al and (zn) determines P uniquely, hence P € M. Since Ai witnesses non-emptiness, we have P f~\ X n M ^ 0. X nA4 is H-set w.r.t. \JU[P n Al]: Let V C \JU[P n Al] be an open cover of X n M such that for all P € P n M we have V n W(P) ^ 0. For every P 6 P \ M select f70(P) £ W(P) such that for all x £ X n M we have 170(P) ^ F(ar). {f/0(P)|P £ P\ A^}UV is an open cover of A" with a dense, finite subcollection W. Then IjWn V is dense in Xn M: Take x &XnM. Since F(x) is an open ultrafilter and (J W is dense there must be a W € W with W € P(z). By construction, W € V and x e dW. Assume now U(Pn Al) ^ X. Then there is P0 £ P such that P 0 nXnA'l = 0. For every P £ P with P n X n M + 0 there is U0(P) £ W(P) with dC/0(P) n P0 = 0. {f/o(P)|P fl X n A4 7^ 0} is an open cover of X n M and has a finite, dense in X n M subcollection {UQ(Pi)\l < n}. Since PI e Ai and W € A4, we have {UQ(Pi)\l < n} £ M and (J{d[/0(P/)|/ < n} e Ai . A contradiction appears because of Example Ib). D Corollary 3. No compact T% space can be partitioned into more than 2N° closed GS sets. D Remark 3. Generalization from 2N° to 2K is possible. Our results specialize to known cardinal theorems if P consists of singleton sets. References [1] Bella, A & F Cammaroto: On the cardinality of Urysohn spaces. Canad. Math. Bull. Vol 31 (2) 1988, 153 - 158. [2] Dow, A: An introduction to applications of elementary submodels to topology. Top. Proc. 13, 1988, 17 - 72. [3] Dow, A & J Porter: Cardinalities of H-closed spaces. Top. Proc. 7, 1982, 27 - 50. [4] Fedeli, A & S Watson: Elementary submodels and cardinal functions. Top. Proc. 20, 1995, 91 - 110.

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[5] Hodel, R E: Cardinal functions I. In: Handbook of Set-theoretic Topology. Eds. K Kunen & J E Vaughan, North-Holland 1984, 1 - 61. [6] Schroder, J: Urysohn cellularity and Urysohn spread. Math. Japonica 38 (6) 1993, 1129 - 1133. [7] Williams, S W: Boxproducts. In: Handbook of Set-theoretic Topology. Eds. K Kunen & J E Vaughan, North-Holland 1984, 169 - 200.

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L-VALUED CATEGORIES: GENERALITIES AND EXAMPLES RELATED TO ALGEBRA AND TOPOLOGY

ALEXANDER P. SOSTAK Department of Mathematics University of Latvia LV-1586 Riga LATVIA E-mail: [email protected] or [email protected] An L-valed category is a category-like conglomerate in which "potential objects" and "potential morphisms" could be respectively objects and morphisms only to a certain degree; this degree is represented by an element of a fixed GL-monoid L — (L, <, A, V, *). In this paper, after brief discussing basic concepts related to L-valued categories, we cosider a series of examples of L-valued categories mainly related to algebra and topology. Mathematics Subject Classifications (2000): Primary 18A05; 3E72; Secondary 20A05; 54A40; 54E05; 54E35. Keywords: L-valued category, fuzzy set, fuzzy group, fuzzy topology, fuzzy uniformity, fuzzy proximity, fuzzy metric.

Introduction The concept of an L-valued category was first introduced in [24], [25] (in a very special case) under the name of a fuzzy category. Later in a series of papers foundations of the theory of L-valued categories were worked out as well as some concrete L-valued categories were considered a. The aim of this work is twofold: first, to give a brief introduction into the basic concepts of the theory of L-valued categories (Section 2), and, second, to present some examples of concrete L-valued categories, mainly related to topology and algebra (Sections 3-7). Some of these categories appeared earlier in [27] and [28]. We start with a short Section 1 devoted to GL-monoids — a concept which is fundamental for our work. a

ln our first papers we used the name a fuzzy category Now we give preference to the term L-valued category or just an L-category as a shorter and more informative 291

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1. Preliminaries: GL-rnonoids To introduce the concept of an L-valued category in a sufficiently general setting and thus to come to natural examples of L-valued categories it is insufficient to restrict to a purely lattice-theoretic context and therefore we enrich the underlying lattice L with an additional algebraic structure. Although different examples empose different demands on these structures, to make exposition homogeneous, we shall assume that the underlying lattice L is enriched with a monoidal operation * in such a way that the resulting quiple (L, <,A, V, *) is a GL-monoid. Although not the most general, this assumption is convenient both to introduce basic notions of the theory of L- valued categories and to expose most of the examples of L-valued categories known to us. Definition 1.1. [11], [12] A GL-monoid is a quiple (L, < V,A,*) where (L, < V, A) is a complete lattice with universal upper and lower bounds 0 and 1 respectively, (0 ^ 1). and * is a commutative monotone binary operation such that: (1) 1 acts as the unity and 0 acts as the zero element: a * 1 = a, a * 0 - 0 Va e L; (2) * is distributive over arbitrary joins, i.e. a * ( V i f t ) = Vi(a*&), Va e L V { f t : i € 1} C L; (3) (L, <, *) is divisible, i.e. a < /3 implies the existence of 7 e L such that a — j3 * 7 It is known that every GL— monoid is residuated, i.e. there exists a further binary operation " i—> " (implication) on L satisfying the following condition: a*/3 < 7 <=> a < (f3 \—> 7)

Va,/3,7£L.

Explicitely the implication is given by e L a*A
(ii) a

__> 13 = i ^ a < __> ( A . 0.) =

Ai(« — > A); (iii) ( V < o < ) ' — » ) 9 = Ai(ai ^^ /?);

fc

(V) a * (A ( ft) = A> * A); (Vi) (a __> 7) * (7 _> 0) <

« ^ fr (vii) a<(3 =>• =>• 7 >— » " < 7 •— > /?•

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Example 1.1. Examples of GL—monoids (1) Heyting algebras. Every complete infinitely distributive lattice (L, <, V, A) viewed as a Heyting algebra (i.e. with * = A) is obviously a GL—monoid. (2) MV—algebras. In the class of GL—monoids MV—algebars can be characterized as those GL—monoids, in which the double negation law holds, i.e. (a i—> 0) i—> 0 = a. (3) Continuous t—norms. Let L = [0,1] be the closed unit interval endowed with the natural order relation <. According to K. Menger (see e.g. the book by B. Schweizer and A. Sklar [21]), a i—norm on [0,1] is a binary operation T : [0,1] x [0,1] -» [0,1] such that (Tl) (T2)

T(x, 1) = x, T(x,0) = 0 (boundary conditions); T(x,y) = T(y,x) (symmetry);

(T3)

T ( x , y ) < T(x',y')

whenever

(isotonosity); (T4) T(x, T(y, z)) = T(T(x,y), z)

x < x' and y < y'

(associativity).

It is known that a triple ([0,1], <, T) is a GL—monoid if and only if the t—norm T is continuous (see e.g. [12, p. 72]). Important examples of t—norms are: Tm(x,y) = x A y (the minimum t-norm) ; T p ( x , y ) = x • y ( the product t-norm ); TL(X, y) = max(:r + y — 1, 0) (Lukasiewicz t-norm) . In the sequel L := (L, <, A, V, *) will always stand for a GL—monoid. 2. L-valued categories: basic concepts 2.1. Definition of an L-valued category An L-valued (or a fuzzy) category C consists of: (1) A class Ob(C) of potential objects. (2) An L-subclass w of Ob(C): u> : Ob(C) —> L. (3) A class M(C] = \J{MC(X, Y) : X, Y e Ob(C)} of pairwise disjoint sets Mc(X,Y) for each pair of potential objects X, Y 6 Ob(C)the members of Mc(X, Y) are called potential morphisms from X toY.

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(4) An L-subclass fj. of M(C): fj, : M(C) —> L, such that if / € MC(X, Y), then /z(/) < w(X) A (5) Composition of morphisms o : Mc(X,Y) x Mc(Y,Z) —> Mc(X, Z) is defined such that the following axioms are satisfied: (i) preservation of morphisms: fj,(g o /) > n(g) * /u(/); (ii) associativity: if / e MC(X,Y), g € Mc(Y,Z) and h £ MC(Z, U), then ho (g o f) = (ho g) o f(iii) existence of identities: for each X € Ob(C) there exists an identity ex € Alc(^) -X") such that fi(ex) = u(X) and for all X,Y,Z € O6(C), all / € .Mc(X,>0 and all g e ,MC(Z, X) it holds / o ex = f and &x ° g — 92.2. Remarks on terminology, notation and comments Remark 2.1. Let Oba(C) = {X € Ob(C) \ w(X) > a}; Ma(C) = {/ € M(C) | M(/) > "}; Mca(X,Y) = {/ e A^c(X,r) | M (/) > a}. The elements of Oba(C) are referred to as a-objects of C, while the elements of Ma(C) are called its a—morphisms. Remark 2.2. Given an L-valued category C = (Ob(C), w, M(C), p,, o) and X € Ob(C), the intuitive meaning of the value w(X) is the degree to which a potential object X of C is indeed its object; similarly, the intuitive meaning of //(/) is the degree to which a potential morphism / of C is indeed its morphism. Remark 2.3. Let C — (Ob(C), M(C), o) be a usual (i.e. crisp) category with Ob(C) as the class of objects and M(C) as the class of morphisms, and o as the composition law. Then C can be viewed also as an L-valued category (Ob(C),u>, M(C),/j,,o) where L-fuzzy classes w : Ob(C) —> L and n '• M(C) —> L are defined, respectively, by the equalities w = I := e and fj, = l. Remark 2.4. Given an L-valued category C = (Ob(C),uj,M(C),^,o) one can construct a crisp category C^ = (Ob(C},Jv[(C),o) by taking all potential objects and potential morphisms as resp. objects and morphisms of CL and leaving the composition law unchanged. The category C x is called the bottom frame of C. This construction can be generalized: let L be an idempotent element of L and let Ct = (ObL(C),Mi.(C),o). Then d is a crisp category having ObL(C) as objects and ML(C) as morphisms. The composition law is the

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same as in C. In particular, Co = CL is the bottom frame and C\ —: C1 is the top frame of C. Of course, the case when CT is empty is not excluded.

2.3. L-subcategories and quotients of L-valued categories Definition 2.1. Let C = (Ob(C),u>, M(C),/J,, o) be an L-valued category. By an (L-valued ) subcategory of C we call an L-valued category C' = (Ob(C),u',M(C),n',o) where u/ < w and /z' < /z. A subcategory C' = (Ob(C),w', M(C), n', o) of an L-fuzzy category C = (Ob(C},w, M(C), ft, o) is called full if /j.'(f) = M (/) A u'(X) A w'(Y) for every / e Thus an L-valued category and its subcategory have the same classes of potential objects and potential morphisms and the same operation of composition. The only difference between them is in L-classes of objects and morphisms, i.e. in the belongness degrees of potential objects and morphisms. Remark 2.5. Let C = (Ob(C),M(C),o) be a crisp category and T> = (Ob(D),M(D),o) be its subcategory. Then T> can be identified with the L-valued subcategory t> = (Ob(C),u',M(C),(i,',°) of C = (Ob(C),w,M(C),n,o), where ^ = 1,^ = 1, such that w'(X) = 1 if X 6 Ob(D) andw'(X) = 0 otherwise; //(/) = 1 if/ € M(D) and/z'(/) = 0 otherwise. In particular, PT = T>. Definition 2.2. An equivalence relation ~ on the class of morphisms of an L-valued category C is called a congruence on C provided that: (1) each ~ —equivalence class is contained in Mc(X,Y) for some X,Y &Ob(C); (2) / ~ /' implies M (/) = /z(/'); (3) if / ~ /' and g ~ g' then g o / ~ g' o f whenever the composition is defined. Given a congruence relation ~ on an L-valued category C, the quotient L-valued category C/ ~ can be defined in a natural way, patterned after the construction of the quotient category in classical situation (cf e.g. [9, p.24]).

2.4. Dilution and concentration of L-valued categories Let C = (Ob(C),u>,M(C),/j,,o) be an L-valued category and, given a natural number n, let the mappings wn : Ob(C) —> L, /z" : M(C) —> L be

296

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defined inductively as follows: * W (X); /x (fe+1) (/) = /(/) * /*(/), * 6 N. Then C" = (O6(C), w", M(C),nn, o) is a L-valued category; it is called the n-ift power of the original L-category C. The operation C —> C" can be interpreted as dilution of the original category C. In case L has square roots (cf e.g. [12]) one can define operation of concentration for L-valued categories. Namely, given an L-valued category C = (Ob(C),w,M(C),u,o) let the L-valued category C1/2 be defined by the equality (0&(C),w 1/2 , -M(C r ),A» 1/2 ,°). The operation C — » C1/2 can be interpreted as the operation of concentration of the original category C. Obviously, crisp categories are invariant under operations of dilution and concentration. 2.5. A general scheme for fuzzification of classical categories Here we describe a general method convenient for fuzzifying classical categories; this method is used in Sections 3 - 7 for constructing concrete L- valued categories. Let C = (Ob(C), M(C), o) and V = (Ob(D), M(D), o) be two ordinary categories and let $ : C —> T> be a functor. We define a new (again) ordinary category Cat by setting Ob(Cat) - Ob(C) and Mcat(X,Y) = MD($(X), $00). Thus the morphisms from X to Y in Cat are the same as the morphisms from $(X) to $(Y) in T>. The composition law in Cat is naturally induced by the composition law in T>. To reflect in notation the way how Cat was obtained we write Cx>* or just CD if it is clear what functor $ is used. Obviously, in case when C = T> and E : C —> C is the identity functor, the resulting category C-DE is just the original category C. Now, defining in a certain way an Z/-subcalss of objects w : Ob(Cx>$) —» L and an L-subclass of morphisms fj, : M(C-p^) —> L satisfying Definition 2.1 we come to an L-valued category (Ob(C-D$),u, M(C-D$),u, o). To denote this L- category we write (Cx>$,u;,/u) 2.6. Functors between L-valued categories Definition 2.3. Let C = (Ob(C),uc,M(C),uc,o) and V = (Ob(D),ujT>,M(D),nv,0) be L-valued categories and let Ft : Ob(C) —> Ob(D) and F2 : M(C) —> M(D) be maps. The quadruple F :=

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(C,P,Fi,F 2 ) is called a functor from C to P ( F : C —> P ) provided the following properties are satisfied: (1) / e AM*,y) implies F 2 (/) e (2) Fa preserves composition, i.e. F2(<7 o /) = F2(<7) o F 2 (/) provided composition g o f is defined; (3) Fa preserves identities, i.e. F2(6x) = &FI(X) f°r anv X (4) Mc (/) < MF 2 (/)) for any / 6 (In [27] a more general concept of a 7-functor was introduced. According to terminology of [27] the functor defined here must be refered to as a 1-functor.)

Remark 2.6. Let C and T> be L- valued categories and let C^ and T>^- be the corresponding bottom frame categories (see Remark 2.4). If F : C —> T> is a functor between L-valued categories then F : CL •—> P1 is a functor in the classic sense. More generally, restricting a functor F : C —» T> to the subcategory CL where i is an idempotent element of L, we obtain a functor FL : d -> VL. 3. L-valued categories of L-sets 3.1. L-valued categories of FL-SET = (L-SET,^,^)

type

Let L-SET be the category of L-(fuzzy) sets, i.e. its objects are pairs: X = (X,A) where X is a set and A : X —> L is its L-(fuzzy) subset, and morphisms / £ M(X, y), where y = (Y, B), are mappings / : X —-> Y such that B o f > A (see e.g. [19], [6]). Given X = (X, A) 6 Ob(L - SET) let wi(A') = inf x A(x) and u2(X) = infx\A(x) V (^4(a;) i—> 0)]. Further, given a morphism / : X —> y in L-SET, let MwX/) = Ui(X) f\Ui(Y),i = 1,2. Thus we obtain L-valued categories F^L - SET = (L - 5FT,Wj,^ Wi ), i = 1,2. Since u>2 > wi we conclude that F\L — SET is a full L-subcategory of F^L — SET. Moreover, in case L is a chain and * = A the both L-categories coincide. Obviously, L - SET = FiL - SET-1, i = 1,2 i.e. L - SET is the bottom frame of these L-categories. The top frame F\L — SET1 in an obvious way can be identified with the category SET of sets. On the other hand, in case when L is an MV-algebra, the top frame of the L-category F2L — SETr can be identified with the category SET2 of pairs of sets (see e.g. [8] for the definition of SET2). In case when we restrict ourselves to consideration of those pairs (X, A) € Ob(L-SET) which have some additional structure, it is possible

298

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to suggest other natural L-valued categories. For example, let L = [0, 1] be the unit interval endowed with some tnorm and let (X, B, m) be a probability space . To define an L- valued category [0, 1}- SET let Ob([Q, I]- SET) consist of all quadruples (X, B, m, A) where (X, B, m) is a probability space and A : X —> [0, 1] is a measurable function. Further, let the morphisms M([Q,1]—SET) be denned in the same way as above. Now, by setting ws(X, B, m, A) = fx A dm we obtain an L-valued category F3[Q, 1}-SET = ([0, l}-

3.2. L-valued categories of FL - SET = (L - SET SET, w, fJ,)-type In the previous subsection, starting from L-SET we defined L-valued categories by assigning to each object X a certain value u}(X) € L which characterizes the measure to which it is "really an object" in the corresponding L-valued category. Here we also start from the category L-SET; however, leaving the class of potential objects unchanged, we extend the class of potential morphisms. To describe this in a formal way we apply the scheme exposed in 2.5. Let C = L-SET, V = SET and let : C -> V be the functor assigning to each X = (X, A) the support set X and leaving morphisms unchanged. Then according to the scheme 2.5 we come to the category L — SET SET', its objects are the same as in L — SET, but its morphisms are all mappings between the corresponding support sets. Starting from this category as the crisp bottom frame we define the L-valued category FL — SET by setting u = e, i.e. u(X) = I for every X & Ob(L - SET) and

for every morphism / from X = (X,A) to y = (Y,B) in the category L — SET SET To show that in the result we obtain an L-valued category, we have to verify that p(g o /) > n(g) * fj.(f) whenever g : Y —> Z, and Z = (Z, C) e Ob(L - SET SET)- Indeed, »(9°f) = l\^x(A >-> C o ( g o f ) ) ( x ) Co (go f))(x)]

> A xex [(A -^ Bof)(x)*(Bof

^

> [A xe x(^ — B o /)(x)] * [A ieX (B o / H-+ C o (g o

Cog)(y)\ = M(/)*M(ff)- Since, obviously, n(ex) = 1, we conclude that FL-

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SET is indeed an L- valued category. The inequality A B o f ( x ) = I for each x e X, i.e. iff ^(/) = 1, and therefore LSET is the top frame of .FL-SET. (The category .FZ^SET is especially visual in case when L= [0,1] and * is the Lukasiewicz conjunction, i.e. it is defined by the equality a * /3 = max{a + /3 — 1, 0}. In this case a i—> /3 — min{l — a + 0, 1} and hence the value l—fj,(f) = max{supx^x(A(x) — (Bof)(x)), 0} can be interpreted as the defectiveness degree of a mapping / from being a morphism in [0,1]-SET (cf e.g. [Subsection 1.8] [23]) )

3.3. L-valued categories of (L — SETREL, u;,//^)- type Let L - SETREL and L - SETREL' denote the categories of L-relations defined as follows. Their objects are the same as in L — SET. Morphisms from X = (X, A) to y = (Y,B) in these categories are L-relations R : X x Y —> L satisfying the inequality

R(x, y) < A(x) A B(y]

\/x£X,y<=Y

in case of the category L — SETREL, and satisfying the inequality V [R(x,y) ^ A(x}}
Vy&Y

in case of the category L - SETREL' (cf [19]). Let X 6 Ob(L - SET) and let Ui(X), i = 1,2 be denned as in Subsection 3.1. Further, given a morphism R : X —> y in L — SETREL let fiui(R) = Ui(X) A uji(y), i = 1,2. In the result we come to L-valued categories (L — SETREL,^,^^), i = 1,2. In a similar way, but starting with the category L — SETREL' we come to L-valued categories (L - SETREL', Ui,^), i = 1,2. The "standard" functor $ : L - SET -» L - SETREL denned by $(A") = A" on objects and $(/) = #/, where Rf(x, y) = ,4(a:), if y = /(x) and Rf(x, y) = 0, if y ^ /(x), on morphisms / : (X, .4) —> (y, B) (cf e.g. [19]), remains also a functor if considered as the functor between L-valued categories: $ : (L - SETM,^)

-» (L -

SETREL,^,^}.

Further, by setting *(#) = X on objects and $(R) — R on morphisms, we obtain the "standard" functor * : L - SETREL -> L - SETREL' (cf [19]), which remains also a functor of L-valued categories: i,tJ:Ui)-+ (L -

300

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3.4. L -valued categories of (L — SETRELL_REL, e, /it) type. Let C = L - SETREL be denned as above and let £> = L - fl£L be the category of sets and L-relations. Its objects are sets and morphisrns from a set X to a set Y are L-relations R : X x Y —> L (recall that the composition of two relations R : X x Y —» L and S : Y x Z —» L is denned by (RoS)(x,z) = \/y&Y(R(x,y) A S ( y , z ) ) . Further, let $ : C -» £> be the functor assigning to (X, A) its support set X e Ob(L - REL) and leaving morphisrns unchanged. Further, let u(X) = 1 for X = (X,A) e ob((L - SETRELL-REL) (= ob(L - SETREL)) and set I*(R) =

/\

(R(x, y) >— » A(x) A B(y))

x£X,y,€Y

for a morphism R : (X,A) i—> (y, J5) in L - SETRELL^REL- An easy verification shows that (L — SETREL,L-REL, e, /u) thus obtained is indeed an L-valued category. Analogously, using the same scheme but starting from the category C = L — SETREL' and setting

M'W = A t( V ^x'2/) A A<x)) t we obtain an L-valued category (L — SETREL' LFunctors $ and \J* introduced above, can be viewed also as functors <E> : (L - SETL-SBT, ^ n) ^ (L - SETRELL-RBL, c, M ); * : (L - SETRELL-REL, e, M) -> (L - S£TfiEL' L _ fiEL , e, M') respectively. To show this it is sufficient to notice that ^(Rf) = M(/) for each mapping / and //(-R) < fJ-(R) for each L-relation R. 4. L-valued categories related to algebra In the previous Section we considered different L-categories whose objects are pairs (X, A) where X is a set and A : X —> L is its L-subset. The subject of this Section are L-categories whose objects are also pairs (X, A), however now X will be supposed to be endowed with some algebraic structure, and this structure will make a certain impact on the conditions which the second component of the pair (X, A) must satisfy. In order to make the exposition here more "homogeneous" we shall restrict to the case when X is a group. Thus, we shall proceed from the category G.R of groups and propose different fuzzifications schemes for GR. Patterned after these schemes one can develop similar fuzzifications

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for other categories of algebra, such as the category RING of rings, the category MOD of modules, the category MON of monoids, the category VECT of vector spaces, etc.

4.1. L-valued categories of (L — GR, u>,^t) type. Let L — GR be the category of L- groups (see e.g. [20] or [19, Chapter 2.3]). Thus, objects of L — GR are L- groups, i.e. triples (X, 0, G) where (X, 0) is a group and G : X —> L is its L-subset closed under operation of product (i.e. G(x Q y) > G(x) * G(y) Vx, y € X) and closed under operation of taking inverses (i.e. G(x~l) > G(x), Vx e X). Given two L-groups X = (X, G) and y — (Y,H), a morphism / : X —> y is a group homomorphism / : X —> Y such that G < H o /. Now we can define fuzzy subclasses o>j : 06(L — GJ?) —> L and /iWi : M(L — GR) —> L by the same formulae as in 3.1 and obtain L-categories

4.2. L-valued categories of (L — G.RGR,U>,^I) type Let L — GRoR be the category defined according to 2.5, i.e. its objects are the same as in L-GR, and its morphisms / : X —> y (x = (X,A), y = (Y,B) e Ob(L - GR)\ are arbitrary homomorphisms / : X -+ Y of the corresponding support groups. By setting w(X) = I and

= A W — » (B ° /)(*)) we obtain an L-valued category (L — GRcR^^). One can modify this category by introducing some "measure" for the class of objects, too. For example, denning Wj, i = 1,2, in the same way as in 4.1 and setting /}$ = H A Hui we come to L-valued categories (L — GRcn, w,, /ij), i = 1, 2

4.3. L-valued categories of (L — GR,u>,n) type In the L-valued categories considered in the previous subsections the class of potential objects was Ob(L - GR) and u; was "estimating" the degree to which X £ Ob(L — GR) belongs to the given L-valued category. Thus, u> in some sense " narrowed" the class of objects if compared with the original category L — GR. On the other hand, here we start with "widening" the class of potential objects from L — GR and then consider an L-valued subclass (jj of this larger class Ob(L - GR) of potential objects. Here are the details:

302

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Let L — GR be the category whose objects are pairs (X, A) where X is a group and A : X —> L is its L-subset (i.e. no validity of axioms from 4.1 on A is assumed) and whose morphisms are arbitrary homomorphisms of the corresponding support groups. By setting (A(x)*A(x')^A(x,x'))}^[/\(A(x-l)^A(x))}, xex

u(X,A) = [ f\ x,x'ex

we define an L-valued class of objects. Now there are several ways how to complete this construction and to obtain a L-valued category. In particular, we may set (J.u(f) = w(X) A (jj(y) for a morphism / : X —> y and thus come to the L-valued category (L — GR,LJ,^). Another possibility is to set #(/) = /u w (/)A//(/) where fj, is defined as in 4.2. Thus we arrive at an Lvalued category (L — GR, u>, p,). Notice that its top frame (L — GR, w, n/ w ) is the category L — GR. By assigning to each (X, A) <E Ob(L - GR) the pair (X, A) e Ob(L SET), i.e. "forgetting" the algebraic structure of X and leaving morphisms unchanged, we define a forgetful functor $ : (L — GR,u},p,u) —> (L - SET SET, e, M) where (L - SET SET, t, M) is the L-valued category denned in 3.2. ^__

^

-r-

5. L-valued categories related to topology

5.1. L-valued categories of FL - TOP = (L- TOP SET, €, /i) type Let L-TOP be the category of Chang-Goguen L-topological spaces [7]. Recall that the objects of L-TOP are L-topological spaces, i.e. pairs (X, T) where X is a set and T is a family of its L-valued subsets (i.e. T c Lx) such that (1) 0,1 e T; (2) U,V & T => U A V £ T and (3) Ui € T Mi € I ==> V^i £ r. The morphisms / : (X,r x ) -> (Y,Ty) in L-TOP are mappings f \X-*Y such that V e TY =» /^OO e T X . Originating from 'the category L-TOP we shall define an L-valued category .FL-TOP as follows. Let L - TOP SET be the category denned according to 2.5, i.e. its objects are L-topological spaces and its morphisms are all mappings between the corresponding support sets. We set w = e. The degree to which a mapping / : X —> Y is a morphism in the L-valued category .FL-TOP is defined by the equality:

= A Very

L-VALUED CATEGORIES

303

where the interior IntA of A € Lx in (X, T) is denned as IntA = \/{U : U e T, U < A}. Since, obviously, //(ex) = 1 for the identical morphism ex '• (X,r) —> (X, r), to ground that FL - TOP = (L - TOP SET, e, /u) thus obtained is indeed a fuzzy category we have to show only that n(g o /) > ^(5) * ^(/) for any mappings f : (X,rx) -+ (Y,Ty), g : (Y,TY) -> (Z,rz). However, this follows from the next chain of (in)equalities (see (vi), (vii) in Section 1.1):

mf, ((g o /^(W) H-» Int((g tof

> mf

inf

jnf jnf

mf (fl-H^)) H->

Int(g-\W)))(y}\*

jnf

Remark (a) The scheme of fuzzification applied here becomes especially lucid when L~ [0, 1] and * is the Lukasiewicz conjunction. Then a i—> /? = min(l — « + /?,!) and hence ^(/) shows how far / deviates from being continuous: //(/) = 1 — d(f] where the value d(f) = supV£TY supx&x(f-l(V) - Int(f-l(V})}(x) can be called the defect of continuity of the mapping / : (X, TX) —> (Y, ry) of the corresponding Chang-Goguen [0,l]-topological spaces. In particular, if (X, TX), (Y, TY) are ordinary topological spaces, then d(f) = 0 if / is continuous and d(f) = 1 otherwise (see [23]). (b) Let I be a continuous lattice [5] and let UJL : TOP —> L - TOP be the embedding omega functor first introduced by R.Lowen [17]

304

SOSTAK

and later extended by T.Kubiak to the case of an arbitrary continuous lattice (see [16]). Then the L-valued category FuL(TOP) obtained by "restriction" of the above fuzzification scheme to the subcategory w L (TOP) of the category L-TOP can be viewed upon as a fuzzification of the category TOP of topological spaces.

5.2.

L-valued categories of FL - FTOP = (L- FTOP SET, e, /x) type

Following [22, 23] by an L-fuzzy topological space we call a pair (X, T) where X is a set and T is an .Muzzy topology on it, i.e. a mapping T : Lx —> L provided with the following properties:

(1) T(0)=T(1) = 1; (2) T(U A V) > T(U) A T(V) (3) T(V 46I t/i) > Ai€z TM

V U, V e Lx; V {Ui : i e 1} C Lx.

A mapping / : X —> Y where (X, Tx), (V, Ty) are .Muzzy topological spaces is called continuous if 7x(/~ 1 (^ / )) > Ty(V) for each V € L y . Let i^FTOP stand for the category of L-fuzzy topological spaces and continuous mappings between them. To define a fuzzification JFL-FTOP of the category L-FTOP we first consider the category L — FTOPsET (cf. Subsection 2.5) whose objects are the same as in L — FTOP and whose morphisms are all mappings between the corresponding support sets. Further, let uj = 1 and let

= A

V€LY

for (X, Tx), (Y, TY) & Ob(L - FTOP) and a mapping f:X-+Y. Thus we come to an L-valued category FL — FTOP = (L-FTOPSET,t,n}- Indeed, if ex : (X,rx) -» (X,rx) is the identity mapping, then obviously, fJ,(ex) = 1. Therefore we have to check only that p,(g o /) > n(g) * /j,(f) for the composition g o / of mappings

L- VA L UED CA TEG DRIES

305

(X,rx) -» (y,7Y) and 5 : (Y,rY) -> (Z,rz) :

/) = A

>[

A 5.3. L-valued categories of FL - FTOP = (L- FTOP SET, w, M Let L — FTOP be the category whose objects are pairs (X, T) where X is a set and T : X —> L is a mapping (no axioms on T are assumed) and whose morphisms from (X, TX) to (Y, 7y) are "continuous like" mappings / : X -> y that is ^(/-^K)) > Ty(V) for every F 6 £y. By setting , T) = (1 — » T(0)) A(l

and

we come to a L-valued category (L — FTOP, w, Taking as the bottom frame category the category L — FTOP SET (cf 2.5) instead of the category L — FTOP and setting

A

eL

we obtain the L-valued category FL - FTOP = (L - FTOP SET, w, //).

306

SOSTAK

6. L-valued categories of proximity spaces Extending somewhat the definition of a fuzzy proximity introduced in [18], see also [28] by an L-fuzzy proximity on a set X we call a mapping 6 : Lx x Lx -> L such that for all A, B, C, A', B' £ Lx: (OP) (IP) (2P) (3P) (4P) (5P)

AA*B;

S(A,B)> [ f\ X

'A The pair (X, 5) is called an L-fuzzy proximity space. A mapping / : (X,6x) —> (Y,Sy) is called proximally continuous, or a proximity mapping, if <5 X (A,S) < 6Y(f(A)J(B))

for all A, J3 e Lx.

Let L-FPROX denote the category of i-fuzzy proximity spaces and proximity mappings. Remark 6.1. Notice that axioms (OP) and (3P) together are equivalent to the following stronger version of the axiom (3P): (3'P) 6(A, BVC) = 6(A,B)V6(A,C).) Remark 6.2. In [18] we restricted to the case when L= [0,1] and * was either the Lukasiewicz conjunction or * = A. Remark 6.3. In [14] and [15] A.K. Katsaras proposed two alternative definitions of fuzzy proximities which can be viewed as special cases of our L-fuzzy proximities. As different from our L-fuzzy proximities, Katsaras fuzzy proximities are essentially two-valued, i.e. they must satisfy the following additional property: (KP)

<5([0,1]*X[0,1] X )C{0,1}.

Besides, to obtain Katsaras fuzzy proximities introduced in [15] we must take L = [0,1] endowed with the Lukasiewicz conjunction. The corresponding category will be denoted K\ — Prox. To characterize Katsaras fuzzy

L- VAL UED CA TEGORIES

307

proximities from [14] we must modify somewhat the above general definition of an L-fuzzy proximity. Namely we must take * = A in the axiom (4P), but define residuation which appear in the axiom (5P) by means of the Lukasiewicz conjunction. The corresponding category will be denoted K2-Prox. In this section we shall discuss some ways in which the category L — FPROX (and, as a by-product, the above mentioned categories of Katsaras fuzzy proximity spaces) can be fuzzified. 6.1.

L-valued categories of (L — FPROX SET, *••> M) type

Let L — FPROX SET be the category obtained from L — FPROX according to the scheme 2.5 and let u> = e. Further, given a mapping / : (X, (Y,6Y),let =

A

To check that (L — FPROXssT, f-, A*) thus obtained is indeed an L-valued category we have to show that /j,(g ° /) > n(g) * /•*(/)• Let / : (X, 5x) —> (Y, 6Y), g '• (Y, Sy) —> (Z, 5z)- Then, applying (vi) from Section 1.1, we get: =

A A,B£

(6X(A,B)^5Y(f(A)J(B)))* X

A

(sY(f(A),f(B))

H-,

5z(g(f(A)),g(f(B))])

A,B&LX

A

(^(C1--0) —^ 8z(g(C),g(D)))

=

C,D^V

6.2. L-valued categories of (L — FPROX, w,/x) type Let now L — FPROX be the category of "proximity-like" spaces, whose objects are pairs (X, 5) where X is a set and 8 : Lx x Lx —> L is a mapping b Note that such estimation of a "potential morphism" in the category L — FPROX SET in case when L = [0,1] is endowed with the Lukasiewicz conjunction earlier appeared in our paper [18].

308

SOSTAK

(no axioms are assumed) and whose morphisms / : (X, 5\) —> (Y,5y) are mappings satisfying the inequality

6X(A,B) < 6Y(f(A)J(B))

for all A, Be Lx .

By replacing the inequality < in an axiom (iP), i=0,l,2,3,4,5 in the definition of the L- fuzzy proximity by the implication sign i—>, one naturally arrives at a fuzzy predicate Ui : LL xL i—> L which can be interpreted as the fuzzy version of the corresponding crisp axiom (iP).c Namely, given a mapping 6 : Lx x Lx —> L let the value w,((5) e L, i = 0,1,2,3,4,5, be defined as follows:

/\{S(A, B) i—> 6 (A1, B') A\(S) = 6(0, 1) i—> 0 (the predicate of universality); (S(A,B) i—> S(B,A))

f\

(the predicate of symmetricity) ;

(6(A, BVC) i—> S(A, B) V 6(A, C]\

(the predicate of union) ;

A,B,C€L ,C€LX

u>4(5) —

A

I A * B i—> S(A, B) }

(the predicate of coincidence) ;

A

/\

Obviously, 5 satisfies a condition (iP), if and only if w»(5) = 1 for the corresponding i = 0,1,2,3,4,5. Starting from the category L-PROX as the bottom crisp frame and basing on the fuzzy predicates Wj, i = 0,l,2,3,4,5, we can define L- valued categories JX-PROX = (L-PROX, w, // w ) where u(X,6) = ^^(6) for (X, 6) e Ob(L - PROX) and ^(f)

= u(X, 8X) A u(Y, SY) for a mapping

°Note that here our ideas have something in common with the ideology of the semantic analysis used by Ying Mingsheng in his theory of fuzzifying topologies, see [29].

L- VALUED CATEGORIES

309

In a similar way but starting from the category L — FPROXsBT (cf Subsection 2.5) instead of the category L — FPROX and setting

P(/) = M/) A M(/) where ^(/) is defined as in Subsection 6.1 we come to the L- valued category

7. L-valued categories related to topological algebra 7.1.

L-valued category PL — TOPGRcR

Let L — TOPGRoR be the category whose objects are quadruples (X, G, ©,r), where (X,Q,G) is an L-group (Section 4.1) and ( X , r ) is an L-topological space (Section 5.1) and whose morphisms are arbitrary homomorphisms between the corresponging support groups ( X , O ) . Given a mapping / : (X,rx) —» (Y,Ty) of L-topological spaces we define its measure of continuity K(/) as in Section 5.1, that is by setting K(/) = f\v^(^€x(f-1(V)(x) >-> Intf~\V)(x))), where Intf-^V) is the interior of the L— set f~l(V) in the L— topology Tx. Further, given a mapping / : X —> Y and L-subsets GX and Gy of X and Y respectively, we define its measure 'j(f) as of a mapping / : (X, GX) —> (Y, GY), by setting

Now, to fuzzify the category L — TOPGRan we set w(X, G,T) =:«(¥>) A «ty) where
-^ are defined, respectively, by ip(x,x') = x Q x' and V(^) = a:"1 for each object (X,G,T) of L - TOPGRcR and = 7(/) A «(/) A u(X, Gx, Tx) A w(X, GY, TY)

for each morphism / : (X,GX,TX) -> (r.Gy.Ty) of L-TOPGRGR. A direct verification shows that /x(ex) = w(X) and /^(y ° /) > ^(5) * //(/) whenever composition ^o/ is defined. Thus we obtain an L-valued category FL - TOPGRaR =(L- TOPGRGR, w, M). 7.2. L-valued category FL — FTOPGRcR Let L — FTOPGRoR be the category, whose objects are quadruples (X, 0, G, T) where (X, 0, G) is the same as above, and T : Lx —» L is an Lfuzzy topology on X 5.2 and whose morphisms are homomorphisms between

310

SOSTAK

corresponding support groups. The fuzzification JFL — FTOPGRoR — (L - FTOPGRGR, w,//) of L - FTOPGRGR is defined similar as in the case of the L-valued category FL — TOPGRoR', the only difference is that now the measure of continuity K of a mapping / : (X,T\] —> (Y, 7y) is defined by «(/) = AV € L- (W) ^ Tx(f-1(V})). References [1] M.Burgin and A. Sostak, Towards the theory of continuity defect and continuity measure for mappings of metric spaces, Acta Univ. Latv.576 (1992), 45 - 62. [2] M.Burgin and A. Sostak, Fuzzification of the theory of continuous functions, Fuzzy Sets and Syst., 62 (1994), 71-81. [3] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), pp. 182-190. [4] D.H. Foster, Fuzzy topological groups J. Math. Anal. Appl, 67, (1979), 549-564. [5] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, A Compendium of Continuous Lattices. Springer Verlag, Berlin, Heidelberg, New York, 1980. [6] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967) 145-174. [7] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl., 43 (1973) 737-742. [8] R. Goldblatt, Topoi: the Categorical Analysis of Logic, - revised ed., North-Holland Publ. Co., Amsterdam, 1981. [9] H. Herrlich, G. Strecker, Category Theory- Berlin, Heldermann Verlag, 1979. [10] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78 (1980) 659-673. [11] U. Hohle, M-valued sets and sheaves over integral commutative dmonoids, In: Applications of Category Theory to Fuzzy Subsets S.E. Rodabaugh, E.P. Klement and U. Hohle eds., Kluwer, Dodrecht, Boston, 1992, pp. 33 - 72. [12] U. Hohle, Commutative residuated 1-monoids, In: Non-Classical Logics and their Applications to Fuzzy Subsets - A Handbook of the Mathematical Foundations of Fuzzy Set Theory, U. Hohle and E.P. Klement eds., Kluwer, Dodrecht, Boston, 1994, pp. 53-106. [13] U. Hohle, A. Sostak, Axiomatics of fixed-basis fuzzy topologies, In: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory , U. Hohle, S.E. Rodabaugh eds. - Handbook Series, vol.3. Kluwer Academic Publisher, Dordrecht, Boston. - 1999. (to appear). [14] A.K. Katsaras, Fuzzy proximity spaces J. Math. Anal. Appl., 68 (1979), 100-110. [15] A.K. Katsaras, On fuzzy proximity spaces J. Math. Anal. Appl., 75 (1980), 571-583.

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[16] T. Kubiak, The topological modification of the L—fuzzy real interval, In: Applications of Category Theory to Fuzzy Subsets, S.E. Rodabaugh, E.P. Klement and U. Hohle eds. - Kluwer Acad. Publ., Dordrecht 1992. - pp. 275 - 306. [17] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl, 56 (1976), 621-633. [18] S. Markin, A. Sostak, Another approach to the concept of a fuzzy proximity, Suppl. Rend. Circ. Matem. Palermo, Ser II, 29 (1992), 530-551. [19] C.V. Negoita, D.A. Ralescu, Application of Fuzzy Sets to System Analysis, John Wiley & Sons, New York, 1975. [20] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. [21] B. Schweitzer, A. Sklar, Probabilistic Metric Spaces, North Holland, Amsterdam, 1983. [22] A. Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser II,, 11 (1985), 89-103. [23] A. Sostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math. Surveys, 44: 6 (1989), 125-186. [24] A. Sostak, Towards the concept of a fuzzy category, Acta Univ. Latviensis (ser. Math.), 562, (1991), 85-94. [25] A. Sostak, On a concept of a fuzzy category, In: 14th Linz Seminar on Fuzzy Set Theory: Non-Classical Logics and Their Applications, Linz, Austria, 1992, 63-66. [26] A. Sostak, Basic structures of fuzzy topology, J. Math. Sci. 78, N 6 (1996), 662-701. [27] A. Sostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, Mathematik-Arbeitspapiere N 48: Categorical Methods in Algebra and Topology (A collection of papers in honor of Horst Herrlich), Hans-E. Porst ed., pp. 407-438. Bremen, August 1997. [28] A. Sostak, Fuzzy categories related to algebra and topology, Tatra Mount. Math. Publ. VbJume 16: Fuzzy Sets: Theory and Applications, Publ. of Mathematical Institute of Slovak Academy of Sciences, Bratislava, 1999. pp. 159-185. [29] Ying Mingsheng, A new approach to fuzzy topology, Part I, Fuzzy Sets and Syst., 39 (1991), 303-321.

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Horst Herrlich's publications A) As Editor 1976

1

Categorical Topology, Mannheim 1975 Lecture Notes Math. 540, Springer 1976 (with E. Binz)

2

Nordwestdeutsches Kategorienseminar, Bremen 1976 Mathematik - Arbeitspapiere 7, Univ. Bremen 1976 (with H.-E. Porst, R.-E. Hoffmann and M.B. Wischnewsky)

3

Categorical Topology, Berlin 1978 Lecture Notes Math. 719, Springer 1979 (with G. Preuss)

4

Structure of Topological Categories, Bremen 1978 Mathematik - Arbeitspapiere 18, Univ. Bremen 1979 (with R.-E. Hoffmann, H.-E. Porst and M.B. Wischnewsky)

1981

5

Special Topics in Topology and Category Theory, Bremen 1981 Mathematik - Arbeitspapiere 25, Univ. Bremen 1981 (with R.-E. Hoffmann, H.-E. Porst and M.B. Wischnewsky)

1984

6

Categorical Topology, Toledo 1983 Heldermann Verlag 1984 (with H.L. Bentley, M. Rajagopalan and H. Wolff)

1985

7

Convergence Structures, Bechyne 1984 Akademie-Verlag, Berlin 1985 (with J. Novak, W. Gahler and J. Mikusinski)

1987

8

Workshop on Category Theory, Bremen 1986 Mathematik-Arbeitspapiere 28, Univ. Bremen 1987 (with R.-E. Hoffmann and H.-E. Porst)

9

Categorical Topology, L'Aquila 1986 Topology and its Applications 27, No. 2 (1987) (with E. Giuli)

1979

313

314

HORST HERRLICH

1989

10

Categorical Methods in Computer Science with Aspects from Topology, Berlin 1988 Lecture Notes Computer Science 393, Springer 1989 (with H. Ehrig, H.-J. Kreowski and G. Preufi)

1991

11

Category Theory at Work, Bremen 1990 Heldermann Verlag 1991 (with H.-E. Porst)

1992

12

Recent Developments of General Topology and its Applications, Berlin 1992 Internat. Conf. in Memory of Felix Hausdorff (1868-1942) Akademie Verlag 1992 (with W. Gahler and G. PreuB)

1999

13

Symposium on Categorical Topology, Cape Town 1994 Univ. Cape Town 1999 (with B. Banaschewski and C. Gilmour)

2000

14

CatMAT 2000 Proc. Conf. Cat. Methods in Algebra and Topology Mathematik-Arbeitspapiere Nr. 54, Universitat Bremen 2000 (with H.-E. Porst)

B) Mathematical Biography 1988

1

Categorical Topology - The Complete Work of Graciela Salicrup Aportaciones Matematicas 2, Soc. Matem. Mexicana 1988 (with C. Prieto)

C) Books 1968

1

Topologische Reflexionen und Coreflexionen Springer Lecture Notes Math. 78 (1968)

PUBLICATIONS

1973

2

Category Theory Allyn and Bacon 1973 2. ed. Heldermann Verlag 1979 (with G.E. Strecker)

1986

3

Einfuhrung in die Topologie: Metrische Raume Heldermann Verlag, Berlin 1986 (with H. Bargenda and C. Trompelt)

4

Topologie I: Topologische Raume Heldermann Verlag, Berlin 1986 (with H. Bargenda)

1988

5

Topologie II: Uniforme Raume Heldermann Verlag, Berlin 1988

1990

6

Abstract and Concrete Categories Wiley-Interscience Publ., New York 1990 (with G.E. Strecker and J. Adamek)

D) Papers 1962

1

Ordnungsfahigkeit topologischer Raume Thesis, Preie Universitat Berlin 1962

1965

2

Ordnungsfahigkeit zusammenhangender Raume Fund. Math. 57 (1965), 305-311

3

Ordnungsfahigkeit total-diskontinuierlicher Raume Math. Annalen 159 (1965), 77-80

4

Ty-Abgeschlossenheit und T^-Minimalitat Math. Zeitschrift 88 (1965), 285-294

5

Wann sind alle stetigen Abbildungen in Y konstant? Math. Zeitschrift 90 (1965), 152-154

6

£-kompakte Raume Habilitationsschrift, Preie Universitat Berlin 1965

7

Nicht alle T2-minimalen Raume sind von 2. Kategorie Math. Zeitschrift 91 (1966), 185

1966

315

316

1967

1968

1969

HORST HERRLICH

8

Quotienten geordneter Raume und Folgenkonvergenz Fund. Math. 61 (1967), 79-81

9

Fortsetzbarkeit stetiger Abbildungen und Kompaktheitsgrad topologischer Raume Math. Zeitschrift 96 (1967), 64-72

10

£-kompakte Raume Math. Zeitschrift 96 (1967), 228-255

11

Properties which are closely related to compactness Indag. Math. 29 (1967), 524-529 (with J. van der Slot)

12

Mengen reeller Zahlen. Ordnungstheoretische und topologische Kennzeichnung Prace Math. 11 (1968), 205-212

13

On intersections of compact sets Duke Math. J. 35 (1968), 439-440

14

ff-closed spaces and reflective subcategories Math. Annalen 117 (1968), 302-309 (with G.E. Strecker)

15

Strengthening Alexander's subbase theorem Duke Math. J. 35 (1968), 671-676 (with G.E. Strecker, E. Wattel and J. de Groot)

16

Compactness as an operator Compositio Math. 21 (1969), 349-375 (with J. de Groot, G.E. Strecker and E. Wattel)

17

Separation axioms and direct limits Canad. Math. Bull. 12 (1969), 337-338

18

Limit operators and topological coreflections Trans. Amer. Math. Soc. 146 (1969), 302-310

19

Topological coreflections Int. Symp. Topology, Herceg Novi 1968 (Belgrad 1969), 187-188

PUBLICATIONS

1970

1971

1972

1974

317

20

On the concept of reflections in general topology Int. Symp. Extension Theory of Top. Structures and Appl., Berlin 1967, (Berlin 1969), 105-114

21

An example in category theory Math. Zeitschrift 113 (1970), 309-312

22

Factorization of morphisms / : B —> A Math. Zeitschrift 114 (1970), 180-186

23

Regular-closed, Urysohn-closed, and completely Hausdorff-closed spaces Proc. Amer. Math. Soc. 26 (1970), 695-698

24

Coreflective subcategories Trans. Amer. Math. Soc. 157 (1971), 205-226 (with G.E. Strecker)

25

A characterization of fc-ary algebraic categories Manuscripty Math. 4 (1971), 227-284

26

Categorical topology Gen. Topol. Appl. 1 (1971), 1-15

27

Algebra n Topology = Compactness Gen. Topol. Appl. 1 (1971), 283-287 (with G.E. Strecker)

28

Coreflective subcategories in general topology Fund. Math. 73 (1972), 199-218 (with G.E. Strecker)

29

A generalization of perfect maps 3. Prague Top. Sym., 1971, (Prague 1972), 187-191

30

Identities in categories Canad. Math. Bull. 15(2) (1972), 297-299 (with C.M. Ringel)

31

Perfect subcategories and factorizations Keszthely 1972, (Coll. Math. Soc. G. Bolyai, Top. 1974), 387-403

318

1975

1976

HORST HERRLICH

32

Regular categories and regular functors Canad. J. Math. 26 (1974), 709-720

33

Topological functors Gen. Topol. Appl. 4 (1974), 125-142

34

Cartesian closed topological categories Math. Coll. Univ. Cape Town 9 (1974), 1-16

35

A concept of nearness Gen. Topol. Appl. 4 (1974), 191-212

36

On the extendibility of continuous functions Gen. Topol. Appl. 4 (1974), 213-215

37

Topological structures Amsterdam Math Centre Tracts 52 (1974), 59-122

38

Topological structures Int. Congr. Math., Vancouver 1974, (Vol. 2, 1975), 63-66

39

Epireflective subcategories of Top need to be cowellpowered Comment. Math. Univ. Carolinae 16 (1975), 713-716

40

Initial completions Math. Zeitschrift 150 (1976), 101-110

41

On the relations P(X xY)=PX xPY Gen. Topol. Appl. 6 (1976), 37-43 (with W.W. Comfort)

42

Extensions of topological spaces Topology (eds. Franklin and Smith Thomas) (Marcel Dekker, New York 1976), 129-184 (with H.L. Bentley)

43

Convenient categories for topologists Topology (eds. Franklin and Smith Thomas) (Marcel Dekker, New York 1976), 71-76

44

Convenient categories for topologists Comment. Math. Univ. Carolinae 17 (1976), 207-227 (with H.L. Bentley and W.A. Robertson)

PUBLICATIONS

1977

1978

1979

45

Some topological theorems which fail to be true Springer Lecture Notes Math. 540 (1976), 265-285

46

Subcategories defined by implications Houston Math. J. 2 (1976), 149-171

47

Products in topology Quaestiones Math. 2 (1977), 45-47 (with H.L. Bentley)

48

The forgetful functor Cont —> Prox is topological Quaestiones Math. 2 (1977), 45-47 (with H.L. Bentley)

49

Cartesian closed topological hulls Proc. Amer. Math. Soc. 62 (1977), 215-232 (with L.D. Nel)

50

The reals and the reals Gen. Topol. Appl. 9 (1978), 221-232 (with H.L. Bentley)

51

Reflective Mac Neille completions of fibre-small categories need not be fibre-small Comment. Math. Univ. Carolinae 19 (1978), 147-149

52

Completion as reflection Comment. Math. Univ. Carolinae 19 (1978), 541-568 (with H.L. Bentley)

53

Completeness is productive Springer Lecture Notes Math. 719 (1979), 13-17 (with H.L. Bentley)

54

Completeness for nearness spaces Amsterdam Math. Centre Tracts 115 (1979), 29-40 (with H.L. Bentley)

55

Semi-universal maps and universal initial completions Pacific J. Math. 82 (1979), 407-428 (with G.E. Strecker)

319

320

HORST HERRLICH

56

Least and largest initial completions, I and II Comment. Math. Univ. Carolinae 20 (1979), 43-77 (with J. Adamek and G.E. Strecker)

57

The structure of initial completions Cahiers Topol. Geom. Diff. 20 (1979), 333-352 (with J. Adamek and G.E. Strecker)

58

Algebra U Topology Springer Lecture Notes Math. 719 (1979), 150-156 (with G.E. Strecker)

59

Initial and final completions Springer Lecture Notes Math. 719 (1979), 137-149

60

Dispersed factorization structures Canad. J. Math. 31 (1979), 1059-1071 (with G. Salicrup and R. Vazquez)

61

Light factorization structures Quaestiones Math. 3 (1979), 189-213 (with G. Salicrup and R. Vazquez)

62

Local compactness and simple extensions of discrete spaces Proc. Amer. Math. Soc. 77 (1979), 421-423 (with V. Kannan and M. Rajagopalan)

1980

63

Equivalence of topologically-algebraic and semitopological functors Canad. J. Math. 32 (1980), 34-39 (with R. Nakagawa, G.E. Strecker and T. Titcomb)

1981

64

Orderability of connected graphs and nearness spaces Amsterdam Math. Centre Tracts 142 (1981), 34-39

65

Topicos de Categorias Topologicas y Algebraicas Univ. Santiago de Chile 1981

66

Universal completions of concrete categories Springer Lecture Notes Math. 915 (1982), 127-135

1982

PUBLICATIONS

1983

67

The coreflective hull of the contigual spaces in the category of merotopic spaces Springer Lecture Notes Math. 915 (1982), 16-26 (with H.L. Bentley)

68

What is algebra? What is topology? Topology, Spec. Seminar Mexico 3 (1982), 16-26

69

Inside Top Durand Math. Assoc. 1982 (Me Master Univ. 1983), 14-21

70

Are there convenient subcategories of Top? Topol. Appl. 15 (1983), 263-271

71

Categorical topology 1971-1981 General Topology and its Relations to Modern Analysis and Algebra 5 Proc. Fifth Prague Topol. Symp. 1981, (Heldermann Verlag 1983), 279-383

1984

72

The quasicategory of quasispaces is illegitimate Archiv Math. 40 (1983), 364-366 (with M. Rajagopalan)

73

Products of quotients in Near Topol. Appl. 17 (1984), 91-99 alias: Y.T. Rhineghost, (with H.L. Bentley)

74

Graciela Salicrup - her mathematical work Categorical Topology (eds. Bentley, Herrlich, Rajagopalan and Wolff) (Heldermann Verlag 1984), 1-22

75

Universal topology Categorical Topology (eds. Bentley, Herrlich, Rajagopalan and Wolff) (Heldermann Verlag 1984), 223-281

321

322

1985

1986

HORST HERRLICH

76

Topological structure theory Convergence Struct. Appl., Schwerin 1983 (Akademie Verlag 1984), 77-80

77

Sequential structures: categorical aspects Convergence Structures (eds. Novak, Gahler, Herrlich and Mikusinski) (Akademie Verlag 1985), 165-176

78

Ascoli's theorem for a class of merotopic spaces Convergence Structures (eds. Novak, Gahler, Herrlich and Mikusinski) (Akademie Verlag 1985), 47-53 (with H.L. Bentley)

79

Filter convergence via sequential convergence Comment. Math. Univ. Carolinae 27 (1986), 69-81 (with R. Beattie and H.P. Butzmann)

80

Cartesian closed categories, quasitopoi, and topological universes Comment. Math. Univ. Carolinae 27 (1986), 235-257 (with J. Adamek)

81

Topological structures and injectivity Rendiconti di Palermo 12 (1986), 87-92

82

Essentially algebraic categories Quaestiones Math. 9 (1986), 245-262

83

Cartesian closed topological hulls as injective hulls Quaestiones Math. 9 (1986), 263-280 (with G.E. Strecker)

84

Concrete categories and injectivity Springer Lecture Notes Comp. Sci. 239 (1986), 42-52 (with H. Bargenda and G.E. Strecker)

85

Galois connections Springer Lecture Notes Comp. Sci. 239 (1986), 122-134 (with M. Husek)

PUBLICATIONS

1987

1988

1989

323

86

The category of Cauchy spaces is cartesian closed Topol. Appl. 27 (1987), 105-112 (with H.L. Bentley and E. Lowen-Colebunders)

87

Topological improvements of categories of structures sets Topol. Appl. 27 (1987), 145-155

88

Factorizations, denseness, separation, and relatively compact objects Topol. Appl. 27 (1987), 157-169 (with G. Salicrup and G.E. Strecker)

89

Kategorientheorie Mathematikstudium? In Bremen? Na klar! Fachbereich Mathematik und Informatik, Univ. Bremen 1987, 21-24

90

Hereditary topological constructs General Topology and its Relations to Analysis and Algebra 6 Proc. Sixth Prague Topol. Symp. 1986 (Heldermann Verlag 1988), 249-262

91

On the representability of partial morphisms in Top and in related constructs Springer Lecture Notes Math. 1348 (1988), 143-153

92

Essentially equational categories Cahiers Topol. Geom. Diff. Categ. 29 (1988), 175-192 (with J. Adamek and J. Rosicky)

93

Sequential structures induced by merotopies Comment. Math. Univ. Carolinae 29 (1988), 679-683

94

Realizations of topologies and closure operators by set systems and neighbourhoods Archivum Math. (Brno) 25 (1989), 83-87

95

Monadic decompositions J. Pure Appl. Algebra 59 (1989), 111-123 (with J. Adamek and W. Tholen)

324

1990

HORST HERRLICH

96

The construct PRO of projection spaces: its internal structure Springer Lecture Notes Comp. Sci. 393 (1989), 286-293 (with H. Ehrig)

97

Functors lifting limits Categorical Topology (eds. Adamek and Mac Lane) World Scientific, Singapore 1989, 89-94

98

Completeness almost implies cocompleteness Categorical Topology (eds. Adamek and Mac Lane) World Scientific, Singapore 1989, 246-256 (with J. Adamek and J. Reiterman)

99

Zero sets and complete regularity for nearness spaces Categorical Topology (eds. Adamek and Mac Lane) World Scientific, Singapore 1989, 446-461 (with H.L. Bentley and R.G. Ori)

100

A characterization of concrete quasitopoi by injectivity J. Pure Appl. Algebra 68 (1990), 1-9 (with J. Adamek)

101 Convergence J. Pure Appl. Algebra 68 (1990), 27-45 (with H.L. Bentley and E. Lowen-Colebunders)

1991

102

Galois connections categorically J. Pure Appl. Algebra 68 (1990), 165-180 (with M. Husek)

103

Remarks on categories of algebras defined by a proper class of operations Quaestiones Math. 13 (1990), 385-393

104

Improving constructions in topology Category Theory at Work (eds. Herrlich and Porst) (Heldermann Verlag 1991), 1-20 (with H.L. Bentley and R. Lowen)

PUBLICATIONS

1992

105

Improving Top: PrTop and PsTop Category Theory at Work (eds. Herrlich and Porst) (Heldermann Verlag 1991), 21-34 (with E. Lowen-Colebunders and F. Schwarz)

106

Algebra U Topology Category Theory at Work (eds. Herrlich and Porst) (Heldermann Verlag 1991), 137-148 (with T. Mossakowski and G.E. Strecker)

107

Convenient topological constructs Applications of Category Theory to Fuzzy Subsets (eds. Rodabaugh, Klement and Hohle) Kluwer Acad. Publ. 1992, 137-151

108

Epireflections which are completions Cahiers Topolog. Geom. Diff. Categ. 33 (1992), 71-93 (with G.C.L. Brummer and E. Giuli)

109

Hyperconvex hulls of metric spaces Topology and Appl. 44 (1992), 181-187

110

Compactness = completeness n total boundedness - a natural example of a non-reflective intersection of reflective subcategories Recent Developments of General Topology and its Applications (eds. Gahler, Herrlich and Preufi) Akademie Verlag 1992, 46-56 (with M. Husek)

1993

111

Categorical topology Recent Progress in General Topology (eds. Husek and van Mill) Elsevier Sci. Publ. 1992, Ch. 11, 369-403 (with M. Husek)

112

Almost reflective subcategories of Top Topology and Appl. 49 (1993), 251-264

325

326

HORST HERRLICH

113

Some open categorical problems in Top Appl. Categ. Struct. 1 (1993), 1-19 (with M. Husek)

114

Compact T0-Spaces and Tb-Compactifications Appl. Categ. Struct. 1 (1993), 111-132

115

On the failure of Birkhoff's Theorem for locally small based equational categories of algebras Cahiers Topol. Geom. Diff. Categ. 34 (1993), 185-192

116

Factorization of flows and completeness of categories Quaestiones Math. 17 (1994), 1-11 (with W. Meyer)

117

Essential extensions of Hausdorff spaces Appl. Categ. Struct. 2 (1994), 101-105

1995

118

A Baire Category Theorem for quasi-metric spaces Indian J. Math. 37 (1995), 27-30 (with H.L. Bentley and W.N. Hunsaker)

1996

119

Compactness and the Axiom of Choice Appl. Categ. Struct. 3 (1995), 1-14

120

An effective construction of a free z-ultrafilter Papers on Gen. Topology and Appl. 11-th Summer Conf. Univ. Southern Maine (eds. S. Andima, R.C. Flagg, G. Itzkovitz, Y. Kong, R. Kopperman, and P. Misra) Annals New York Acad. Sci. 806 (1996), 201-206

121

Categorical Topology - its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 Handbook of the History of General Topology (eds. C.E. Aull and R. Lowen), Kluwer Acad. Publ. Vol. 1 (1997), 255-341 (with G.E. Strecker)

122

When is N Lindelof? Comment. Math. Univ. Carolinae 38 (1997), 553-556 (with G.E. Strecker)

1994

1997

PUBLICATIONS

1998

327

123

Choice principles in elementary topology and analysis Comment. Math. Univ. Carolinae 38 (1997), 545-552

124

The Ascoli Theorem is equivalent to the Boolean Prime Ideal Theorem Rostock Math. Kolloq. 51 (1997), 137-140

125

Stabilitat topologischer Eigenschaften unter kategoriellen Konstruktionen Mathematik aus Berlin (edt. H.G.W. Begehr) Weidler-Verlag Berlin 1997, 245-250

126

The Ascoli Theorem is equivalent to the Axiom of Choice Seminarberichte FB Mathematik Univ. Hagen 62 (1997), 97-100

127

Maximal filters, continuity and choice principles Quaestiones Math. 20 (1997), 697-705 (with J. Steprans)

128

Morita-extensions and nearness-completions Topol. Appl. 82 (1998), 59-65 (with H.L. Bentley)

129

Doitchinov's construct of supertopological spaces is topological Serdica Math. J. 24 (1998), 21-24 (with H.L. Bentley)

130

The historical development of uniform, proximal, and nearness concepts in topology Handbook of the History of General Topology (eds. C.E. Aull and R. Lowen), Kluwer Acad. Publ. Vol. 2 (1998), 577-629 (with H.L. Bentley and M. Husek)

131

Countable choice and pseudometric spaces Topology and Appl. 85 (1998), 153-164 (with H.L. Bentley)

328

1999

2000

HORST HERRLICH

132

Categorical properties of probabilistic convergence spaces Appl. Categ. Struct. 6 (1998), 495-513 (with Dexue Zhang)

133

Compactness and rings of continuous functions - without the axiom of choice Proc. Symp. Cat. Topology, Cape Town 1994, Univ. Cape Town 1999 (eds. B. Banaschewski, C.R.A. Gilmour, and H. Herrlich), 47-54 (with H.L. Bentley)

134

On simultaneously reflective and coreflective subconstructs Proc. Symp. Cat. Topology, Cape Town 1994, Univ. Cape Town 1999 (eds. B. Banaschewski, C.R.A. Gilmour, and H. Herrlich), 121-129 (with R. Lowen)

135

Composing special epimorphisms and retractions Cahiers Topol. Geom. Diff. Cat. 40 (1999), 221-226 (with L. Schroder)

136

Productivity of coreflective classes of topological groups Comment. Math. Univ. Carolinae 40 (1999), 551-560 (with M. Husek)

137

Products, the Baire category theorem, and the axiom of choice Comment. Math. Univ. Carolinea 40 (1999), 771-775 (with K. Keremedis)

138

Weak separatedness for nearness spaces Indian J. Math. 41 (1999), 15-24 (with D. Vaughan)

139

Powers of 2 Notre Dame Journal of Formal Logic 40 (1999), 346-351 (with K. Keremedis)

140

Generating ordered topological spaces from LOTS Topol. Appl. 105 (2000), 231-235 (with E. Kronheimer)

PUBLICATIONS

2001

141

Nearness, subfitness and sequential regularity Appl. Categ. Structures 8 (2000), 67-80 (with A. Putlr)

142

Free adjunctions of morphisms Appl. Categ. Structures 8 (2000), 595-606 (with L. Schroder)

143

Abstract Initiality Comment. Math. Univ. Carolinae 41 (2000), 575-583 (with L. Schroder)

144

The Baire Category Theorem and Choice Topol. Appl. 108 (2000), 157-167 (with K. Keremedis)

145

On countable products of finite Hausdorff spaces Math. Logic Quart. 46 (2000), 537-542 (with K. Keremedis)

146

On the strict completion of a nearness space Quaestiones Math. 24 (2001), 39-50 (with H.L. Bentley)

147

Is the construct of L-topological spaces a co-tower extension of some simpler construct? Quaestiones Math. 24 (2001), 141-149 (with D. Zhang)

148

The functor that wouldn't be Categorical Perspectives (eds. J. Koslowski and A. Melton) Birkhauser, 2001, 29-35

149

The emergence of functors Categorical Perspectives (eds. J. Koslowski and A. Melton) Birkhauser, 2001, 37-45

329

330

2002

HORST HERRLICH

150

The naturals are Lindelof iff Ascoli holds Categorical Perspectives (eds. J. Koslowski and A. Melton) Birkhauser, 2001, 191-196

151

Abelian groups: simultaneously reflective and coreflective subcategories versus modules Categorical Perspectives (eds. J. Koslowski and A. Melton) Birkhauser, 2001, 265-281

152

Free factorizations Appl. Categ. Structures 9 (2001), 571-593 (with L. Schroder)

153

Weak factorization systems and topological functors Appl. Categ. Structures 10 (2002), 237-249 (with J. Adamek, J. Rosicky, and W. Tholen)

154

Products of Lindelof T% spaces are Lindelof - in some models of ZF Comment. Math. Univ. Carolinae 43 (2002), 319-333

155

A characterization of the prime closed filter compactification of a topological space Quaestiones Math. 25 (2002), 381-396

156

On a generalized small-object argument for the injective subcategory problem Cahiers topol. geom. diff. categ. 43 (2002), 83-106 (with J. Adamek, J. Rosicky, and W. Tholen) (with J. Adamek, J. Rosicky, and W. Tholen)

157

Zum Begriff des topologischen Raumes In: Felix Hausdorff: Gesammelte Werke, Bd. II, Springer 2002, 675-744 (with M. Epple, M. Husek, G. Preufi, W. Purkert, and E. Scholz

158

Trennungsaxiome In: Felix Hausdorff: Gesammelte Werke, Bd. II, 745-751 (with M. Husek and G. Preufi)

PUBLICATIONS

331

159

Zusammenhang In: Felix Hausdorff: Gesammelte Werke, Bd. II, 752-756 (with M. Husek and G. Preufi)

160

Abzahlbarkeitsaxiome In: Felix Hausdorff: Gesammelte Werke, Bd. II, 757-761 (with M. Husek and G. Preufi)

161

Hausdorff-Metriken und Hyperraume In: Felix Hausdorff: Gesammelte Werke, Bd. II, 762-766 (with M. Husek and G. Preufi)

162

Vervollstandigung und totale Beschranktheit In: Felix Hausdorff: Gesammelte Werke, Bd. II, 767-772 (with M. Husek and G. Preufi)

163

Striking differences between ZF and ZF+ weak choice in view of metric spaces Quaestiones Math. 25 (2002), 405-420 (with K. Keremedis and E. Tachtsis)

164

The Boolean prime ideal theorem holds iff maximal open niters exist Cahiers topol. geom. diff. categ. 43 (2002), 313-315 alias Y.T. Rhineghost (with K. Keremedis and E. Tachtsis)

165

Injective hulls are not natural Algebra Universalis 48 (2002) 379-388 (with J. Adamek, J. Rosicky, and W. Tholen)

2003

166

The axiom of choice holds iff maximal closed niters exist Math. Logic Quarterly 49 (2003), 323-324

2004

167

Merotopological spaces Appl. Categ. Struct. 12 (2004) 155-180 (with H. L. Bentley)

168

On closure operators, the reals, and choice (with E. Giuli) [to appear]

332

HORST HERRLICH

169

Zur Existenz maximaler Filter und Ideale [to appear]

170

On the epireflective hull of Top in Near (with H. L. Bentley and J. Carlson) [to appear]

Fur HH, von BB, nach CM Palmstrom mochte gerne horen wie das Unendliche zu denken sei: dass Mengen der Zahl eins, zwei, drei und soweiter, dazu nicht gehoren steht natiirlich vollig fest? und drum unendlich sicher ist alles was jene nach sich lasst. Aber wie ist nun das Leere zu verstehn? Palmstrom kann hier nicht umhin auch einen Hauch Unendlichens zu sehn? jedoch v. Korf doziert mit Strenge: Nein, "nichts" ist eine endliche Menge. Tertium Non Datur Korf und Palmstro kamen zu erwagen ob und wie das ausgeschloss'ne Dritte zu begriinden sei. Sollte wohl das Zweierlei ein vollig unanriittelbarer Zug jedweden Wissens sein, oder ist das schlichte "Ja-vel-Nein" doch durch subtilere Prinzipien zu ersetzen? Korf lehnt eher erst'rem zu, diese Richtung vorgewiesen glaubend durch manche wichtige Autoritat: siehe Aristoteles und Boole. Palmstrom ist dem weniger gewogen, ahnt formalen Kurzschluss hier, verstossend gegen konstruktive Wirklichkeit, und er formuliert die Bitte: ach, gewahrt mir doch das Dritte.

333

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