Introduction to Uniform Spaces

Lxjndon Mathematical Society Lecture Note Series. 144

Introduction to Uniform Spaces I.M. James Savilian Professor of Gcomclry Malhemalical Insiiiuie, Uiiiversily of Oxford

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C A M B R n X i E UNIVERSITY PRESS Cambridge New York

Port Chester Melbourne

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011, USA 10, Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1990 First published 1990 Printed in Great Britain at the University Press, Cambridge Library of Congress cataloguing in publication data available British Library cataloguing in publication data available

ISBN 0 521 38620 9

(.1

Contents

Introduction 1.

Uniform

structures

2.

I n d u c e d and c o i n d u c e d u n i f o r m

3.

The u n i f o r m

4.

Completeness

5.

Topological

6.

Uniform

transformation

7.

Uniform

s p a c e s over a base

8.

Uniform c o v e r i n g Append i x :

topology completion

groups

spaces

f iIter s

Exercises Bibliography Index

and

groups

structures

iniroauciion

T h i s b o o k i s b a s e d on a c o u r s e given at

the University

and g r a d u a t e

students.

but t h e remainder i s published

of Oxford

Here f o r

and o t h e r s

in

subject

in Chapter

at l e a s t

an o u t l i n e

II

of

Bourbaki

text

several

s p e c i a l i z e d monographs,

Roelcke

and D i e r o l f

to general

many f i n d

chapters

topological

stage.

no k n o w l e d g e o f

topology.

b a s i c k n o w l e d g e of

topology

u n i f o r m w o r l d and t h e

Γ 2 ].

it

is

[25]

the

However,

on t o p o l o g y

such as I s b e l l

the

contain are

[ 7 J, P a g e

[19J,

others. the transition is

from

metric

a major s t e p

Uniform

I have t h e r e f o r e

i n s u c h a way t h a t

a c c o u n t of

Moreover t h e r e

spaces

extremely d i f f i c u l t .

intermediate

books

amongst

For s t u d e n t s o f m a t h e m a t i c s spaces

classical

I n d e e d much o f

The c l a s s i c

the theory.

[21],

undergraduati

s p a c e s was d e v e l o p e d by W e i l

t h e more r e c e n t of

recently

time.

the t h i r t i e s .

of

lectures

the m a t e r i a l i s

not so well-known.

the f i r s t

majority

sixteen

t o an a u d i e n c e o f

About h a l f

The t h e o r y o f u n i f o r m

is

of

which

s p a c e s make an

w r i t t e n the

excellent

first

two

t h e y c a n b e r e a d by a s t u d e n t The s e c o n d two c h a p t e r s

with

assume a

and a r e aimed a t s h o w i n g how t h e

t o p o l o g i c a l world are

related.

The m o n o g r a p h s on t h e t h e o r y m e n t i o n e d a b o v e are w r i t t e n tr.ainly

w i t h t h e n e e d s of

analysts

in mind.

Rather

o v e r t h e same g r o u n d a g a i n I h a v e c h o s e n t o e x p l o r e aspect

of

the s t a r t

the theory. that

Although

it

t h a n go

a different

has been r e c o g n i z e d

tODoloaical q r o u p s c a n p r o f i t a b l y

be

from

regarded

as unLXorm s p a c e s , that

it

is

I do n o t b e l i e v e

possible

groups.

t o d e v e l o p a t h e o r y of uniform

on n a t u r a l l y

to the

in the f i n a l

presented

t h e o r y of u n i f o r m

a b a s e and h e n c e t o t h e t h e o r y o f

uniform

transformation here.

spaces

covering

over

forty

find

exercises,

a l s o an a p p e n d i x c o n t a i n i n g

a short bibliography

t2J.

an o u t l i n e

book I h a v e ,

[21 J.

Ling for

[ 2 4 J.

and P r o f e s s o r some h e l p f u l

final

draft

e s c a p e d my

of

Finally,

the r e s u l t s

the

concerning

greatly relied

of Arens [ I J , like

on

by t h e work

of

consulted

Collins

t o t h a n k Mr.

Don S h i m a m o t o , w h o a t t e n d e d t h e

suggestions;

is

text.

of c o u r s e ,

I would

literature

There

Of t h e many p a p e r s I h a v e

I would p a r t i c u l a r l y m e n t i o n t h o s e and d e V r i e s

of

A l s o I h a v e b e e n much i n f l u e n c e d

R o e l c k e and D i e r o l f

and

mainly derived xron t h e

w h i c h a r e n e e d e d i n t h e main

In w r i t i n g t h i s Bourbaki

over

spaces,

on t h e s u b j e c t b u t w i t h some new p r o b l e m s a s w e l l .

filters

appreciated

section.

At t h e e n d t h e r e a d e r w i l l of

has been l u i i y

An o u t l i n e o f s u c h a t h e o r y i s

This leads

a set

it

Γ4] Paul

lectures,

l a t t e r very kindly

read

t h e t e x t and d e t e c t e d v a r i o u s e r r o r s w h i c h

the had

attention.

Mathematical I n s t i t u t e , J u l y 1989

Oxford

1.

Uniform structures

Let u s s t a r t

by r e c a l l i n g

from

t h e t h e o r y of r e l a t i o n s .

set

X

is

We w r i t e to

just ζΗη

η.

a subset when

Given

ξ

R

(ζ,η)

some n o t a t i o n

and

terminology

Formally a r e l a t i o n of

the c a r t e s i a n

square

and s a y t h a t

ξ

is

R-related

For e a c h s u b s e t

Η

of

€ R

we d e n o t e

on a g i v e n X χ X.

by

R[5] = {η : ζΚη} the s e t

of

R-relatives

of

ζ.

X

we

write R[H] = Note t h a t

υ RCei CcH

if

.

{Hj} i s

a family

of

s u b s e t s of

X

then

R[uHj ] = u R [ H j ] ; in g e n e r a l ,

however,

R[nH^ ]

The i d e n t i t y r e l a t i o n X X X.

A relation

reflexiv e. given

The r e v e r s e

on

a proper

X

is

nR[H^J.

j u s t the diagonal

which c o n t a i n s of

s u b s e t of

ΔΧ

a relation

R

is is

said

to

ΔΧ

of

be

the r e l a t i o n

R^

by if

We d e s c r i b e

R

and o n l y i f as symmetric

identity relation

is

The c o n p o s i t i o n is

R

is

the r e l a t i o n

ηΚξ

.

if

R = R ^;

for example

the

symmetric. of r e l a t i o n s

R <> S

on

X

R,

g i v e n by

S

on t h e ζ(Η

same s e t

» S)n

if

X

and onl'i

i-

I.M. James

if

ς5ζ

and

Also note

ζΗη

associative

ξ

Composition of

(but g e n e r a l l y

not commutative)

is

Η

for repeated compositions

length

such t h a t

i = 0 , . . . , n - l

equivalence

the

sarjection

.

to

η

ζ

when

i.e.

the n a t u r a l After of u n i f o r m

R

x^ = η

and t r a n s i t i v e and

R[ζ J

π : X + X/R,

»

.

is

bracketing

as

related

to

η

by

of

x^Rx^^^

points

for

if

R = R c R.

then

R

is called classes

is

if

is

s a i d t o be an

the equivalence

i s d e n o t e d by

g i v e n by

R

π(ξ)

= R[5J ,

X/R is

class and

called

projection. these preliminaries structure

The members o f the n o t a t i o n s

on a g i v e n

Ω on

for

all

This

such

the

definition

structure certain

relations

t h e o r y are s t i l l

A uniform X χ X

X.

are not c a l l e d

relation

(I . 1 ) .

set

square X χ χ s a t i s f y i n g

the f i l t e r of

we a r e r e a d y f o r

but

s t r u c t u r e on a g i v e n

set

that

ΔΧ c D

D t Ω ,

(ii)

D e Ω

implies

D ^ e Ω ,

D e n

implies

Ε « Ε c D

for

some

Ε e Ω.

is

a

conditions. (although

used)

(i)

(iiil

and s o

such

is

and

as t r a n s i t i v e

on t h e c a r t e s i a n

Definition

relations

a sequence

The s e t o f e q u i v a l e n c e

a filter

=

.

relation,

ς.

factors)

n,

symmetric

of

(n

x^ = ζ ,

We d e s c r i b e reflexive,

of

R"-related

an R - c h a i n o f

filter

Note t h a t

X.

i s unnecessary

X

ς.

= R[S[H]]

for e a c h s u b s e t

of

seme

that

(RoS)[H]

Thus

for

entourages.

X

is

uniform Λΐ! uLiuf c^ Note t n a t that

for

any

E" c D. is

to

(iii) η > 1

exists

this

to the

an e n t o u r a g e

condition Ε

such

that

e x t e n s i o n w i t h o u t comment i n what

follow.

uniform

structure

notation. uniform

on

structure

refine s

fi,

Ω = Ω'

is

Definition X an

s p a c e we mean a s e t X;

usually

Λ r e f i n e m e n t of

an e n t o u r a g e o f

is

there

We s h a l l u s e

By a u n i f o r m

set

e x t e n d s by i t e r a t i o n

Ώ'

X

alone

a uniform

In t h i s

is

sufficient

structure

Ω

coarsens

The d i s c r e t e

i s the structure

Ω of

is

a

Ω

is

s i t u a t i o n we s a y t h a t β'.

t o be e x c l u d e d we d e s c r i b e

(1.2).

together with a

such t h a t each entourage

Ω'.

or t h a t

X

If

Ω'

the p o s s i b i l i t y

the refinement

uniform

also

structure

as

that

strict.

on a g i v e n

i n which e v e r y s u p e r s e t of

the

diagonal

entourage. In t h i s

space.

s i t u a t i o n we d e s c r i b e

Clearly

o t h e r uniform Definition

the d i s c r e t e

structure.

(1.3).

set

X

sole

entourage. In t h i s

space.

uniform

filter

uniform

points

structure

uniform

s u p e r s e t s of

that the base g e n e r a t e s the

X

every

X

X χ χ

is

the

uniform

i s refined

has a t

by

least

and t h e

three

well.

so t h a t t h e members of the base,

In t h e c a s e

of

two

trivial

least

structures as

t h e members o f filter.

on a g i v e n

has at

structure

When

by a b a s e ,

set

structure

Provided

are d i f f e r e n t .

is given

uniform

refines

as a t r i v i a l

t h e r e are other uniform

a filter are the

X

uniform

structure.

the d i s c r e t e

structure

distinct If

points

structure

i n which the f u l l

the t r i v i a l

e v e r y o t h e r uniform

as a d i s c r e t e

At t h e o t h e r e x t r e m e we h a v e

s i t u a t i o n we d e s c r i b e

Clearly

distinct

uniform

The t r i v i a l

i s the structure

X

the

we s a y a uniform

I.M. James s t r u c t u r e we s a y t h a t t h e b a s e g e n e r a t e s and d e s c r i b e t h e members o f

t h e uniform

the base as b a s i c

structure

entourages.

For e x a m p l e t h e s y m m e t r i c e n t o u r a g e s a l w a y s form a b a s e , we s h a l l be u s i n g

frequently.

i s g e n e r a t e d by t h e Reversing for

a filter

our v i e w p o i n t , Ω

(1.1), (i)

as

on

X χ X.

s u p p o s e t h a t we have a b a s e For

ii

for

all

implies

Ε c

(iii)

D e 8

implies

Ε » Ε c D

Apart from t h e m o d i f i c a t i o n

for

to

For e x a m p l e ,

consider

d i t i o n s are s a t i s f i e d = { (ζ ,η) ,

sane

for

(ii),

a r e t h e same a s t h o s e on

Η χ Ε

Ω

some

therefore,

the

conditions

itself.

the r e a l

line

by t h e f a m i l y

of

JR.

The t h r e e

|ζ - η| <

:

where

e

runs through the p o s i t i v e

reals.

structure

defined

i n t h e c a s e of that

for a l l

JR. B",

e Χ.

on a s e t

ρ

Β

v a n i s h e s on ξ , η e Χ.

X

Euclidean

follows.

i s a non-negative

satisfying ΔΧ.

The

s t r u c t u r e can be

or more g e n e r a l l y a s

ρ : X x X

for a l l

S Ρ(ζ,ζ)

ε,η,ζ

A s i m i l a r uniform

a pseudcmetric

First

= Ρ(η,ζ).

Ρ(ξ ,η)

on

function

as follows.

con-

subsets

uniform

ρ(ς,η)

those

Ε e B,

s t r u c t u r e g e n e r a t e d by t h i s b a s e i s c a l l e d t h e

real-valued

to

Ε £ Β ,

uniform

Recall

structure

D e Β ,

D e Β

of

β

follows.

ΔΧ c D

6

t o be a u n i f o r m

three conditions corresponding

(ii)

on

structure

diagonal.

the base has t o s a t i s f y in

The d i s c r e t e u n i f o r m

a fact

three

conditions,

Secondly,

Finally

+ Ρ(ζ,η) Α pseudcmetric

ρ

determines

a uniform

umjorm airuciurcs s t r u c t u r e on

X

by t a k i n g

U^ = { (ζ , η ) of

Χ χ Χ,

3

: ρ (ζ ,η)

where

ε

For e x a m p l e

as base t h e

family

runs through t h e p o s i t i v e pseudometric,

and p o s i t i v e o f f

the diagonal,

determines

structure,

the t r i v i a l

where,

determines the

uniform

structure.

determined by

by

ρ

is

uniform

reals.

which i s

constant

the d i s c r e t e

pseudometric,

trivial

Of c o u r s e d i f f e r e n t

subsets

< c)

the d i s c r e t e

while

of

which i s

uniform

zero every-

structure.

p s e u d c m e t r i c s may d e t e r m i n e

t h e same

For e x a m p l e t h e u n i f o r m

structure

the

structure

same a s t h e u n i f o r m

determined

2p . Observe t l i a t

pseudometric

ρ

of

of

the family

positive

any uniform

in this

fashion

entourages

rationals.

since

it

Suppose t h a t

is

uniform

structure

D , η

with

define

a real-valued

α (5,η)

= 2

(ξ,η)

σ(ζ,η)

function

σ(ξ,η)

ρ : X χ X -<· ®

a base of

for

an i n t e g e r

all

η

{D,^)

n. so

later.

for

Next we

that

Finally,

the

by

)} k>o

where t h e summation r a n g e s o v e r c h a i n s k

with If

ζ = x^ X

is

and

x^^ , . . . ,χ^^

of

length

η = Xj. .

a uniform

space the i n t e r s e c t i o n

R

of

the

entouragei

that

such

can

countable

symmetric

otherwise.

i s defined

=

with a

the

the

we s h a l l do

structure

σ : X x X ->• ]R

= 0

runs through

a l s o t r u e , as

anything

D^^, c D n+1 η

consisting

which I a n i t

consisting

there e x i s t s

e ^n^'^n+l

pseudonetric

and

c

is

of

inductively

(n=0,l,...)

D„ = X X X 0

if

the converse

has a uniform

Then we c a n c o n s t r u c t

d e t e r m i n e d by a

where

construction,

base.

X

admits a countable base,

not r e q u i r e d for X

on

ρ ^[0,e),

In f a c t

be shown by t h e f o l l o w i n g details

structure

the

6

IM. James

entourages c o n s t i t u t e s is clearly reflexive let

D

E.

Since

an e q u i v a l e n c e

and s y m m e t r i c .

be a n y e n t o u r a g e .

This i s

R C Ε

true for

equivalence

Then

relation,

D as

R

For

some

entourage

hence

R c R.

R

transitivity,

for

R •> R C Ε = E,

and s o

X.

To e s t a b l i s h Ε » Ε C D

we h a v e t h a t

every

r e l a t i o n on

Thus

R " R c D. R

is

an

asserted.

/ / *

β ill When R

of

Basic

e n t o u r a g e d e t e r m i n e d by

X

a finite

is

the entourages

constitutes uniform

relations,

X.

(1.4). if

the diagonal

R

entourages.

For e x a m p l e , pseudcmetric ρ ^(0)

= Δ,

p, i.e.

if

and t h e s e

Returning

A uniform

separated the

case,

Δ

intersection

which

therefore

In f a c t

to the

in turn correspond

to the general

coincides

with the

and o n l y i f

structure is

is

set

X

is

intersection

i s d e t e r m i n e d by a

separated ρ

precisely

c a s e we make

on a g i v e n

the uniform

the

equivalence

structure

the structure if

the

structure.

correspond p r e c i s e l y

fcn t h e p a r t i t i o n s

of

an e n t o u r a g e

the uniform

in the f i n i t e

Definition

in p a r t i c u l a r ,

is itself

a base for

structures

set,

partition.

if

and o n l y

a metric.

if

Umjorm structures The c l a s s

of

η

separated uniform

in p r a c t i c e .

Two o t h e r c l a s s e s

important,

are

as

Definition

(1.5).

X

The u n i f o r m

such t h a t

Of c o u r s e i n t h e c a s e of generated

D

of

it

always t o t a l l y

is

only t o t a l l y

is

this

space

take in

D

from

D,

(1.5)

by2

is

Definition if

for

is

related

if

subset

S

trivial

is

term.

uniform

uniform

a finite better

if

subsets,

for

each entourage

D

S^

structure set. us

of

D

X

Μ x Μ <= D. by a

a finite

structure

let

Μ

c a n be c o v e r e d

Then

of

X.

X

bounded.

totally

Uj^

of

X

subset

X.

the

finite

by a m e t r i c

is

of

is

i s determined

t h a t no p o i n t o f

is

G i v e n an e n t o u r a g e

X

structure

satisfied

structure

that

t h e d i a m e t e r of

+ diamSj^

a finite

important c o n d i t i o n

m u s t be bounded i f

implies

the

or simply D - s m a l l ,

seme p o i n t o f

finite

also

bounded

t h e c o n d i t i o n t o be

itself

t o be t h e b a s i c e n t o u r a g e

(1.5)

totally

l e t us say t h a t a s u b s e t

When t h e u n i f o r m ρ

X

technical X

number o f D - s m a l l

then

is

while the d i s c r e t e

bounded when

a n a l l of o r d e r

X

for

For e x a m p l e ,

a further

in

X

there e x i s t s

sufficient

bounded,

To u n d e r s t a n d

condition

X

space

b a s i c e n t o u r a g e s when t h e u n i f o r m

is

t h e uniform

s p a c e w h i c h are

DCS] = X .

by a b a s e .

introduce

of uniform

importance

follows.

for each e n t o u r a g e of

s p a c e s i s of g r e a t

The

ρ

on

For

condition

i s d i s t a n c e more than 1 S^^

is defined

of

X.

Since

and t h e n

ρ

is

S^

is

bounded

.

(1.6).

The u n i f o r m

every entourage

D

of

space X,

X

i s uniformly

every pair

connected

of

p o i n t s of

ξ,η

there

X

by a D - c h a i n .

In o t h e r w o r d s ,

for

e a c h p a i r of

points

exists

V

an i n t e g e r if

k

such that

the condition

is

(ξ,η)

satisfied

e D .

for

Of c o u r s e i t

is

sufficieni

b a s i c e n t o u r a g e s when t h e

8

ι Μ. James

uniform line

structure

®

i s uniformly

more g e n e r a l l y line Δ

®

= Δ

i s generated

so i s

for

all

η

i s never uniformly one p o i n t .

connected, the real

i s uniformly

n-space

connected,

we s e e

connected,provided

defined

(1.7).

as

the

sense,

than

structure

is

structure-preserving are the uniformly

φ : X -»• Y,

X

whenever

Ε

is

X

where

if

X

(φ χ φ)

an e n t o u r a g e o f

i s automatically uniformly

of

structure

s p a c e h a s more

s p a c e s , i s uniformly continuous

φ

Since

con-

follows.

The f u n c t i o n

an e n t o u r a g e o f Thus

uniform

t r i v i a l uniform

spaces

inVerse-image

tinuous functions,

are uniform

rational

same s t r u c t u r e .

the

real

structure,

Also the

with the

the

connected.

in the

Definit ion

E".

that the d i s c r e t e

In t h e t h e o r y of uniform functions,

For e x a m p l e ,

with the Euclidean

On t h e o t h e r hand t h e

always uniformly

structure

by a b a s e .

^E

Y is

Y.

continuous if

i s d i s c r e t e or t h e uniform

and

structure

the

uniform

of

Y

is

trivial. Clearly

the i d e n t i t y function

uniformly

continuous.

uniformly

continuous,

then the composition Thus u n i f o n n functions

Also if where

as morphisms.

c a l l e d uniform Clearly

φ : X ->· Y

X, Y

ψφ : X ->• Ζ

s p a c e s form

on any uniform

and

Ζ

it

is

for

all

structure

of

Y

Ζ

The e q u i v a l e n c e s

of

are

spaces,

continuous.

with the uniformly

sufficient

for

continuous

the category

the condition

basic entourages i s generated

E,

when t h e

by a b a s e .

s t r u c t u r e s a r e d e t e r m i n e d by p s e u d o m e t r i c s

Proposition

ψ : Y

are uniform

i s uniformly

a category,

is

are

equivalences.

satisfied

be e x p r e s s e d

and

space

as

in

(1.7)

to

be

uniform

When t h e u n i f o r m the condition

can

follows.

(1.8).

Let

X

and

Y

be u n i f o r m

s p a c e s where

the

umjorm siruciurea uniform

structures

respectively. uous i f ρ(ζ,η)

for

This,

a r e d e t e r m i n e d by p s e u d c m e t r i c s

Then a f u n c t i o n

each

< δ

e > 0

implies

real

fore

uniform

is

functions

equivalences.

is

(1.9).

surjection,

where

Ε

earlier

Let X

an e n t o u r a g e o f X

of

such that

X

c ELφS],

totally

bounded,

A similar

X,

are

Y

to

are m e t r i c

all

spaces

continuous,

bijections,

are

there-

transformations

of

under

invariant. invariant

The

three

in t h i s

and

the

sense. last

as

(1.10).

surjection,

where

spaces.

If

X

is

φ

of

Then

i s uniformly

Then is

Y.

D = (φχφ)

^E

continuous.

exists a finite φ(0[8])

finite,

this

subset

= φΧ = Y.

Ξ

Since

shows t h a t

Y

is

as

follows

asserted.

X

connected

continuous

Y.

since

φΞ

be a u n i f o r m l y

are uniform

bounded t h e r e

result

holds for

uniform

Let

φ : X -»• Y

and

Y

connectedness,

be a u n i f o r m l y

are uniform

then so i s

which p r o v i d e s

spaces.

continuous If

X

is

Y.

T h i s can e a s i l y be e s t a b l i s h e d our n e x t r e s u l t ,

all

φ : X -»· Y

D [ S ] = X.

Proposition

uniformly

Y

which h e l p the behaviour of

and

totally

φ(015])

ξ ,η e Χ.

are uniformly

a uniform

be any e n t o u r a g e

Since

all

contin-

that

spaces which i s i n v a r i a n t

t o t a l l y bounded then s o i s

is

and

σ,

understood.

Proposition

let

such

which i s f a m i l i a r

The o r t h o g o n a l

called

sane r e s u l t s

t w o t o be b e t t e r

and

iscmetries.

p r o p e r t i e s we d i s c u s s e d

is

for

which are d i s t a n c e - p r e s e r v i n g

equivalence

For

X

ρ

i s uniformly

δ > 0

< ε,

When

A p r o p e r t y of uniform

We now p r o v e

a

the c o n d i t i o n

analysis.

a r e e x a m p l e s of

uniform

: X ->• Y

there e x i s t s

the distance-preserving Isometries,

φ

σ(φ(ξ),φ(η))

of c o u r s e ,

students of

j

d i r e c t l y ot

a useful

as a c o r o l l a r y

characterization.

of

J ,ιγι. Jumci Proposition if

(1.11).

and o n l y i f

uniformly

for

The u n i f o r m each d i s c r e t e

continuous

function

For s u p p o s e t h a t uniformly

the preimage

We h a v e

d'^ = D,

ς,η

then

and s o

λ(ξ)

(ξ,η)

D = {-1,+1}

define

for

and

i

D α (λχλ)

Δ

not constant

are

still

this

are uniform

D

of

X

λ

is

D'^

λ : X

JD

λ(x)

= +1 λ

I)

is

the diagonal

is

discrete,

Δ

Thus

of

D.

if

therefore

(ξ,η)

e D

constant.

D

of

connected.

X

and a

for

all

k .

Taking

by

λ(χ)

= -1

when

otherwise.

the

Given a

not uniformly

i s uniformly

completes

every

where

n,

connected

constant.

δ" = Δ.

sane

X

( ξ , η) /

(ζ,χ)

pair

ε D^

Then

continuous.

Since

λ

is

proof*

forming

a c a t e g o r y where

s p a c e s but the morphisms are

(1.12) •

Y

of

Consequently

in the d i r e c t

Definition

^Δ

D

connected.

: X ->• D ,

since

for

a n o t h e r way of

uniform

preserving

« D",

and s o

There i s

n,

is

a symmetric entourage

such t h a t

some

λ

suppose that

Then t h e r e e x i s t s e X

all

= λ(η).

Conversely

ς,η

i s uniformly

is uniformly

space

: X -»· Ε

D = (λχλ)

for

X

uniform

λ

continuous function

consider

ε X

X

space

image s e n s e ,

The f u n c t i o n

as

i s uniformly

there e x i s t s

an e n t o u r a g e

Y,

open i f Ε

objects

structure-

follows.

φ : X

spaces,

the

of

where

X

for each Y

such

and

entourage that

Ε[φ (χ) ] c φ ( D C x ] ) for

all

X £ X.

Of c o u r s e i t for

sufficent

if

the condition i s

b a s i c e n t o u r a g e s when t h e u n i f o r m

by a b a s e . when *

is

Y

For example

is discrete.

φ

is

structure

of

always uniformly

In the s i t u a t i o n

of

(1.8)

T h e r e is an error in t h e corresDOndino result

satisfied X

is

given

open the (9.34)

condition of

[101.

for

φ

exists

t o be u n i f o r m l y an

η € Y,

e > 0

implies

open i s

such t h a t that

that

for

σ(φ{χ),η)

ρ(χ,ζ)

< δ for

each

< t,

some

δ > 0

there

where ξ 6 X

χ e X such

and

that

Φ (ξ.) = η . Note that its inverse continuous uniform

i s uniformly

(1.13).

continuous.

Let

and l e t

are uniform

and o n l y

Consequently

open i f

if

a uniformly

and o n l y i f

it

is

a

φ : X

ψ ; Y

spaces.

Ζ

If

Y

be a u n i f o r m l y

be a f u n c t i o n ,

ψφ : X

Ζ

continuous

where

X,

i s uniformly

Y

and

open

then

ψ . For l e t

is

open i f

equivalence.

surjection

so i s

i s uniformly

b i j e c t i o n i s uniformly

Proposition

Ζ

a bijection

Ε

be any e n t o u r a g e o f

an e n t o u r a g e o f

exists

X.

an e n t o u r a g e

If

F

of

ψφ

Y.

Then

i s uniformly

Ζ

such

is

uniformly

D = (φχφ) open t h e n

^E

there

that

F[ψφ (x) J c ψφΟ[χ] for

all

X £ X.

Then

ρ[ψ (y) ] c ψΕΓγ:; for

all

y £ Y,

Proposition ψ : Y ->· Ζ and

Ζ

so i s

and s o

(1.14).

ψ

Let

φ : X

be a u n i f o r m l y

are uniform

continuous

spaces.

If

be a f u n c t i o n injection,

ψφ

and

where

i s uniformly

let X, Y

open

then

φ . For l e t

D

be a n y e n t o u r a g e o f

open t h e r e e x i s t s an e n t o u r a g e Ε[ψφ(χ)] is

Y

open.

c ψφθ[χ]

an e n t o u r a g e o f

Suppose t h a t

for

all

Y ,

η £ Ε[φ(χ)]

F

points

since .

ψ Then

of

X . Ζ

If such

χ £ X .

is

uniformli

that

Now

i s uniformly ψ(η)

ψφ

Ε =

(ψχψ)~^Ε

continuous.

ε Γ[ψφ(χ)]

c ψφΟ[χ]

I.M. James and s o

ψ(η)

=

ψφ(ξ)

i n j e c t i v e we h a v e Ε[φ(x)]

c φΟ[χ]

Proposition where

so i s

some

η = φ(ζ) and s o

(1.15).

X, Y

uniformly

for

and

Ζ

φ

: X

open s u r j e c t i o n .

Since

η e φΟ[χ]

Y

ψφ

is

Thus asserted.

ψ : Y

spaces.

If

.

open, as

and

ψ

Ζ

be

functions,

Suppose t h a t

i s uniformly

φ

is a

continuous

then

ψ .

an e n t o u r a g e of Since

ψ

φ (ζ)

Ζ

Ε[φ(χ)]

e Ε[φ(χ)]

η £ D[x] this

and t h e n

Proposition function,

then

X Y

For s i n c e

for ζ

Let and

X e X

χ £ X .

then

So

Since

continuous,

φ : X

Y

for

an e n t o u r a g e

ψφ(ζ)

as

Ε

X .

of

Y

some = ψφ(η)

open

If φ

X

is

and

of

of

asserted.

spaces.

X

is

Y

is

surjective.

Φ such

is

uni-

that

= ΦΧ , Hence

η=1,2,... (φ{χ),η)

for

be a u n i f o r m l y

a r e uniform

Ε

F

if

φ (ξ) = φ ( η )

i s an e n t o u r a g e of

χ £ Χ .

we h a v e

£ X

Y

open t h e r e e x i s t s

Ε"[φΧ] c φχ

i s an e n t o u r a g e

i s uniformly connected then

X x X

points

all

i s uniformly

Είφ (χ) ] = φ( ( Χ χ Χ ) [ χ ] ) all

If

o p e n t h e r e e x i s t s an e n t o u r a g e

seme

(1.16).

n o n - e m p t y and

D = (ψφχψφ)"^Ρ

(ψφ (x) ,ψφ (η) ) £ F .

ψ

where

i s uniformly continuous.

c φο[χ]

for

shows t h a t

formly

ψφ

i s uniformly

such that

as

.

i s uniformly

are uniform

For s u p p o s e t h a t

for

and s o

φ

Let

ζ e D[x]

Ε [ φ χ ] c φχ

.

e e"

So i f for

seme

and

η £ Υ η

so then for

and s o

any

η £ φΧ

required. To c o n c l u d e

this

section

p l a y s an i m p o r t a n t r o l e mentioned pair

of

in the

transverse

X χ X ,

to each other i f

the diaqonal

ΔΧ .

concept

which

in the theory although I cannot find

literature.

s u b s e t s of

I introduce a further

F i r s t of a l l , for their

l e t us say that a

a given set intersection

More s p e c i f i c a l l v ,

it

X ,

are

i s contained

suppose t h a t

X

is

in a

Uniform structures uniform

s p a c e and t h a t

Let us

R

i s an e q u i v a l e n c e r e l a t i o n

s a y t h a t an e n t o u r a g e

D η R = ΔΧ . determined that φ

l:

R = (φχφ)~^ΔΥ than

entourage

is

,

φ : X

Y ,

R .

Note t h a t

of

s i n c e any s u b s e t

also

we s i m p l y

transverse.

X

to the

and o n l y

(1.17) .

where

Y

X ,

if

is

Let

and

Ζ

i s a uniformly

verse

to

ψ

then

(ii)

if

φ

i s a uniformly

then if

and

Y

φ Y

is

is

Φ(ζ)

e D

and

= φ{η)

,

one

to

transverse of

trans-

entourage

structure of

X

is

is

is

to the projection

continuous

of

to

to

ψ

where

Ε

hence X

X

(iii),

transverse

to

φ

and l e t

transverse

to

ψ ;

then

transverse

D Ε

Y

is

X

is

is

trans

transverse

to

which i s

transverse

transverse

to

to

ψφ .

i n t h e c a s e of to

(φ ( ξ ) , Φ ( η ) )

ψ;

ψφ ;

e Ε

and

let

D

then

E,

transverse

be a n e n t o u r a g e o f

(i),

if

(ii) ,

be an e n t o u r a g e of

D η (φχφ)~^Ε

t o ψφ .

is

Thus,

transverse

of

let

X

In t h e c a s e of

which i s

Y

ψφ ;

if

then

then

ζ = η.

i s an e n t o u r a g e

i n j e c t i o n and

is transverse

ψφ (ξ ) = ψ φ ( η )

be f u n c t i o n

ψ ;

continuous, to

Ζ

Then:

o p e n s u r j e c t i o n and

transverse

,

ψ : Y

spaces.

transverse

I n t h e c a s e of

which i s

so

transverse

consisting

and

are s t r a i g h t f o r w a r d .

a n e n t o u r a g e of (1.12),

is

transverse

D = (φχφ)~^Ε

(ξ,η)

in

X

i s uniformly

The p r o o f s let

of

structure

is

is a set, is

the uniform

Y

a r e uniform

φ

φ

Y

a transverse

transverse

φ : X

If

(iii)

of

say t h a t

it

(i)

ΨΦ

R

X .

point-space.

Proposition

to

t e r m when

D

a base

For e x a m p l e t h e u n i f o r m

if

if

where

to

the e x i s t e n c e

verse entourages, transverse;

R

and t h e n we s a y t h a t

implies the e x i s t e n c e

discrete

transverse

We s h a l l g e n e r a l l y u s e t h i s

by a f u n c t i o n

rather

D

on

X Y

to

so be as ψ .

which which

i s an e n t o u r a g e of

is is X

It

i.ivi.jomes By a l o c a l u n i f o r m

and u n i f o r m l y uniform verse

open f u n c t i o n

spaces,

to

.

The r e a s o n f o r

equivalences

Let

continuous functions,

Y is

are trans-

emerge

in

local uniform

By c o m b i n i n g

we o b t a i n

φ : X -»• Y

where

X

local

i s a l o c a l uniform e q u i v a l e n c e .

continuous

and

is also a

A l s o t h e c o m p o s i t i o n of

(1.18).

X

s t r u c t u r e of

equivalence

w i t h our p r e v i o u s r e s u l t s

Proposition

where

the terminology w i l l

Clearly a uniform

uniform e q u i v a l e n c e .

(1.17)

φ : X + Y ,

such t h a t t h e uniform

φ .

Section 3

e q u i v a l e n c e we mean a u n i f o r m l y

X, Y

and

and

Ζ

ψ : Y ->• Ζ

be u n i f o r m l y

are uniform

spaces.

Then: (i)

if

φ

i s an i n j e c t i o n ,

a l o c a l uniform

equivalence,

ψφ

then

i s uniformly ψφ

is

o p e n and

ψ

a l o c a l uniform

is

equiva-

lence; (ii)

if

uniform

i s a uniformly

φ

equivalence then

One f u r t h e r Proposition function, φ

(1.19).

where

φ

φ : X ->• Y Y

inverse

ψ

e n t o u r a g e of

Y

such t h a t

continuous. Υ . since

If Ε'

(Γΐ,φψ(η)) Ji(n)

is

η c Ε'[φ(χ)]

ε Ε ο Ε ,

= ΨΦΨ(η)

Ε ο Ε

Ε

is t r a n s v e r s e

to

an e n t o u r a g e of

and

Suppose

and l e t

Consider the entourage

f- F ,

continuous that

equivalence.

open.

b e a n y e n t o u r a g e of

^E

spaces.

is

which i s a l o c a l uniform

D

(φψχφψ)

local

equivalence .

be a u n i f o r m l y

a r e uniform

For l e t

F =

is a

w h i c h may be w o r t h m e n t i o n i n g

and

i s uniformly

and ψφ

i s a l o c a l uniform

ψ

Let

X

admits a l e f t

Then

result

open s u r j e c t i o n

,

where

X

Y ,

since

symmetric

ψ . is

Then

uniformly

Ε' = Ε η F η (ψχψ)~^0

χ e Χ ,

η « Ε[φ (χ) ] ,

φψ

be a

then

since

which i s

transverse

we o b t a i n t h a t

η = ΦΦ (η)

φψ(η) £ Ε [ φ ( χ ) ] ,

Ε ' c: Ε .

to .

ψ.

of

Hence

Since

But we a l s o

have

Uniform structures ψ (n)

e D[x]

,

since

η = ΦΨ (η)

£ ψΟ[χ].

uniformly

open,

uniformly

continuous

of

Y

for

this

i

(ψχφ)Ε'

c D ,

Therefore

as asserted.

Ε'[φ ( χ ) ]

so c φΟ[χ]

I n f a c t we o n l y

and t r a n s v e r s e

conclusion

and

t o be

need

t o the uniform

obtained.

and s o ψ

φ t o be

structure

is

2.

Induced and coinduced uniform

S u p p o s e t h a t we h a v e a f u n c t i o n a set

and

Y

(φχφ)

^E,

where

uniform

is

a uniform Ε

structure

φ

trivial

on

X,

if

Y

called

continuous.

structure

X.

structure

then

relation

determined

More g e n e r a l l y

Y

d e t e r m i n e d by t h e p s e u d c m e t r i c structure

σ

then

φ : X •>· Y

and

uniform

space

and

uniform

structure

is

transitive

ψ : Y X

and

Ζ Y

induced

by

X

are s e t s .

φ.

Or we c a n g i v e

structure

induced

by

ψφ;

the r e s u l t

X

uniform

is clearly

structure

. sense.

Ζ

X

the

^ΔΥ

uniform

We c a n g i v e

by

is Y

a the

the uniform uniform

the

same.

of

the

induced

ψ : Y

Ζ

property i s characteristic

structure.

Proposition where

has the

where

and t h e n g i v e

the

example,

in the following

induced

uniform

has

R = (φχφ)

σ(φχφ)

structure

The f o l l o w i n g

Y

has the

be f u n c t i o n s ,

ψ

for

has t h e uniform

d e t e r m i n e d by t h e p s e u d c m e t r i c

The a b o v e p r o c e d u r e Let

if

if

X

by t h e e q u i v a l e n c e

φ.

structure.

For another

structure generated

is

Y, g e n e r a t e a

structure

For e x a m p l e ,

then so does

X

images

the induced uniform

has the d i s c r e t e uniform

by

where

The i n v e r s e

as the c o a r s e s t uniform

i s uniformly

uniform

φ : X -»• γ ,

runs through the e n t o u r a g e s of

T h i s may b e d e s c r i b e d which

space.

structures

X,

(2.1). Y

the uniform i s uniformly

and

Let Ζ

structure

φ : X ->• Y

and

are uniform

spaces.

i n d u c e d by

ψ

continuous

if

Suppose t h a t

from t h a t o f

and o n l y i f

ψφ

be

is

Z.

uniformly

functions, Y Then

has φ

Induced and coinduced uniform structures continuous. For l e t for

Ε

be a b a s i c

some e n t o u r a g e

structure. is

If

ψφ

an e n t o u r a g e

of

(φχφ)~^Ε =

iently

uniform

i s uniformly

for

φ : X

Λ useful

special

φ

continuous

is

X

and

then

(ψφχψφ)

continuous,

is

^F

injective

as is

t o be u s e d . Y

injective

continuous

sufficient Let

X

and

(uniformly

X

suffic-

We d e s c r i b e

are uniform

and

asserted.

spaces,

has t h e

as

induced

injections

condition

φ : X ->• Y Y

i s given

left

embedd-

in

be a u n i f o r m l y

are uniform

continuous)

are unifonn

spaces.

continuous

Suppose t h a t

inverse.

Then

(>

is

is

an

embedding.

For i f

ψ : Y -»• X

e n t o u r a g e of

X

e n t o u r a g e of

Y .

then

is a left (φχφ)

The i n d u c e d u n i f o r m

structure

structure

is called

the r e l a t i v e uniform

is called t r a c e s on

to a subset

a s u b s p a c e of A χ A

φ : X ·<- Y,

embedding

if

and o n l y

a uniform

equivalence;

r e l a t i v e uniform

of

where

A

where

is

the

of

structure, X .

and

Y

structure.

φΧ ,

φ

and

D

Ε = (ψχψ)

^D

is

s t a n d a r d way of

and

space A,

The e n t o u r a g e s X .

are uniform

the function here

of

a uniform

t h e e n t o u r a g e s of X

if

inverse

^E = D

a uniform

ion

has the induced uniform

terminology

where

if

(2.2).

where

admits a

a uniform

the

= Ε ,

structure.

function,

ure,

Then

Consequently

Y ,

embedding

Proposition

φ

Y

X and s o φ i s u n i f o r m l y

Not a l l u n i f o r m l y ings.

since

c a s e when t h e f u n c t i o n

important

a uniform

Z,

X ,

of

special

a function

of

Y .

(φχφ)(ψχψ)

i s an e n t o u r a g e The

F

entourage of

giving

X ;

this

with t h i s of

A

are

struct just

Note that a functspaces,

i s a uniform

X + φΧ d e t e r m i n e d by of c o u r s e ,

an

i s given

the

φ

is

I.M. James

Trace of

Proposition uniform

(2.3).

space

each index

j. D

full ξ ί

set X

seme

space

X

meets

Xj^

for

Then

Let

by u n i f o r m l y all

seme

j

totally

X. is

X^

Then

such t h a t a finite

j,

and s o

{Xj}

of

the

bounded

for

D^ = D|(X^xX^)

totally

is

and

covering

bounded.

X^

of

χ e S

(2.4).

for

is

totally

S = uS^

ς e Xj

Proposition

X^

Since S^

Then

X e S^.

is

X^ .

subset

Xj.

then

X

subspace.

be a f i n i t e

be an e n t o u r a g e o f of

a finite

{X^}

Suppose t h a t Then

an e n t o u r a g e

exists

Let

X.

For l e t is

an e n t o u r a g e on a

(χ,ξ)

(χ,ζ) as

be a c o v e r i n g subspaces.

and some

k.

Then

D^LS^]

s u b s e t of

£ D,

connected

bounded

there is

the

X.

e D^

If

for

required.

of

the

uniform

Suppose t h a t X

is

x^

uniformly

connected. For l e t discrete. index Let

j,

φ : X Then

D

φ|Χ^

be u n i f o r m l y i s uniformly

and s o c o n s t a n t ,

since

X^

continuous, continuous,

where for

i s uniformly

a. be t h e c o n s t a n t v a l u e o f φΐχ.. 1 3 each pair of i n d i c e s i , k such t h a t X^

Then meets

D

is

each connected.

a . = a, , for J X,. . Therefore

inuuccu anu Lutnuucea umjorm sirucmres Φ

itself

is constant,

Subspaces of bounded.

totally

In f a c t

Proposition Y

then so i s

is

For l e t

D

of

Y

E'

E'[T] = Y

for

such t h a t y^ £ T* since

be t h e E'Cy^]

E'

is

S

E·

If

that

Y.

Let

sane f i n i t e

subset

subset consisting meets the

s e t of

Then

is

X

A

of

of

of

those

φ.

there e x i s t s

points

For e a c h

^E

entourage

t o t a l l y bounded

Then

y^

we Y.

of

Τ

point

y^ £ Ε ' [ φ ( χ ^ ) ]

so c Ε[φ(χ_;_)].

x^'s.

X

is

D = (φ*φ)

Τ =

x^ £

° Ε·[φ{χ^}]

X

bounded

be a s y m m e t r i c Y

image of

totally

X .

E'

Since

is

Then

X = D[S]

space

i s uniformly

i s uniformly an e n t o u r a g e

special

open.

of

required.

open,

open i f Ε

as

X

be g i v e n t o t h o s e w h i c h a r e u n i f o r m l y

the subspace

totally

structure.

of

» E' c E.

the i n c l u s i o n function

of

spaces are a l s o

Y

Among t h e s u b s p a c e s o f a u n i f o r m should

connected,

be a f u n c t i o n where

entourage

s y m m e t r i c and

be t h e

Y

space.

of

choose a point

E'Ly^] Let

Ε

such t h a t

T' c τ

φ : X

be a b a s i c

seme e n t o u r a g e

Let

Let

i n the induced uniform

for

have

i s uniformly

bounded u n i f o r m

a uniform

X ,

X

we h a v e

(2.5).

a s e t and

and s o

ly

in the

sense

Specifically,

for X

attention

each

such

entourage

that

E[x] c A η D[x] for

all

points

χ £ A ,

Of c o u r s e

set satisfy

this condition.

Proposition

(2.6).

α , 3 : X ->• Y be t r a n s v e r s e

Let

be u n i f o r m l y to

Y .

t h e empty s e t and t h e

full

We p r o v e

X, Y

and

Ζ

be u n i f o r m

continuous functions If

φα = φ3

spaces.

Let

and l e t

φ : Y -»• Ζ

tlien t h e

coincidenci

I.M. James set

Μ = Μ(α,Β)

of

For

be

let

transverse a,Β

Ε

to

α

β

i s uniformly

open i n

a symmetric e n t o u r a g e of

φ .

Let

are uniformly

are entourages

and

D

be any e n t o u r a g e

continuous

of

X .

Y

the preimages

X .

such t h a t

of

X .

Ε <> Ε

Since

( α χ α ) ( β χ β )

Write

D' = D η (αχα)~^Ε η (βχβ)~^Ε If

ζ

α(χ)

c D'[u(x)] = Β(χ)

and

( α ( ζ ) , β (ξ))

where

to

φ ,

for

= φβ(ξ)

= β(ζ),

χ ε Μ ,

(2.6)

with

(1.16)

Let

X, Y

(2.7).

uniformly

connected.

functions

and l e t

φα = φ β

Let

and

,

then Therefore

and

i.e.

ξ

and s o

Μ

we Ζ

α, β : X

φ : Y

Ζ

Ε · Ε

ε Μ . is

is

Thus

uniformly

obtain be uniform Y

set

spaces,

be u n i f o r m l y

be t r a n s v e r s e

then the coincidence

to

with

X

continuous

Y .

Μ = Μ(α,β)

is

either

full.

The n o t i o n extended

α(ς)

φα(ς)

all

Corollary

empty or

u : Μ c χ

X .

By c o m b i n i n g

If

and

£ Ε, (β ( ζ ) , β (χ) ) e Ε .

However

hence

C Μ η D[x]

open i n

χ £ Μ

(α(ζ),α(χ))

£ Ε · Ε .

transverse C/tuix)]

,

of

induced uniform

to multiple

example of

this

of u n i f o t m

spaces.

with a family

procedure.

{iTj}

π . : nx , . The u n i f o r m

Specifically

of

projections,

course,

product

let

The c a r t e s i a n p r o d u c t

s t r u c t u r e on

which each of

uous.

We r e f e r

to

uniform

product,

and w r i t e

We h a v e t o

The u n i f o r m

of

{x^}

ΠΧ^

is

be

an

be a f a m i l y

cones

equipped

where

X. .

product

structure for

situations.

structure can,

ΠΧj,

show t h a t

Πχ^

is

the coarsest

these projections with t h i s

uniform

uniform

i s uniformly structure,

contin-

as

the

IIX^ = X . such a s t r u c t u r e e x i s t s .

For

this

Induced and coinduced uniform structures p u r p o s e we i d e n t i f y lI(Xj X X j ) cartesian

X x X

with the cartesian

by t h e o b v i o u s r e a r r a n g e m e n t of product

llDj

,

where

where

is

Dj

and w h e r e j. is

all

the image of

Dj .

Thus

also uniformly

TTj

i s not

index

of

Xj

for

under

each

HD^ ,

index

number o f

j

indices

the p r o j e c t i o n

only uniformly

j

product

cartesian products

but a f i n i t e

ΠΟ^

each

the

π^ χ

continuous

but

open.

In p r a c t i c e explicit

for

for

Then

The u n i f o r m

of

through the entourages

Dj = X^ χ X^

Note t h a t just

X χ Χ .

g e n e r a t e d by t h e f a m i l y

runs

factors.

D^ c x^ χ χ^

can be r e g a r d e d as a s u b s e t of structure

product

it

is

description

seldcm n e c e s s a r y

of

one p r o c e e d s by t a k i n g

the uniform

to refer

product

a d v a n t a g e of

back t o

the

structure.

the following

Instead

characteristic

property. Proposition Let

φ : A

and

nx^

(2.8). ΓΚ^

is

tinuous if

and o n l y i f

indexing

Δ

set,

Δ

and i t

left

is

of

Δ : X -I- X X X

let

ρ

of uniform is

φ

spaces.

a uniform

space

i s uniformly

the functions

For e x a m p l e ,

follows

φ^ = π^φ

con-

: A ->- Χ^

π^

for

X^ = X

power,

a t o n c e frctn

,

take

where

(2.8)

continuous. any

j,

that

J the

for

all

is

the

diagonal

In f a c t

since

we o b t a i n frcra

(2.2)

embedding.

product i s d i s c r e t e

family. is

J*^^

i s uniformly

of d i s c r e t e uniform

X

A

Then

the uniform

a uniform

an i n f i n i t e

Let

each of

inverse

The u n i f o r m family

where

product.

obvious.

Δ : X ->- X"^

has the

that

is

nXj = x"^,

function

be a f a m i l y

continuous.

The p r o o f Then

{Xj}

be a f u n c t i o n ,

t h e uniform

i s uniformly

j.

Let

spaces but generally

Note t h a t

uniformly

be a uniform

be a p s e u d o m e t r i c

in the case

open i f

space, on

the diagonal

X.

and o n l y i f

w i t h uniform It

of

a

finite

not in the

case

function X

is

structure

discrete. Ω,

i s a simple e x e r c i s e to

and show

I.M. James that the distance if

and o n l y i f

defined that

by

ρ

Ω

p.

is

Thus

Ωρ

Ω

where

Ω^

ρ

e n t o u r a g e of

section is

with r e s p e c t

for

Ω^.

Ω.

structure

Ωρ

structure

.

of u n i f o r m

For l e t

Then t h e r e e x i s t s such t h a t

a

D

such

Clearly

Ω

is

continuous.

by t h e c o l l e c t i o n

ρ

on

and

of

previous

such t h a t

a refinement Moreover

are

sequence

D^^ = D

X

of

{D } η

Ω^,

D e Ω Ρ

such uniform

struct-

be a

As we h a v e s e e n i n t h e

a pseudcmetric

^ i s uniformly

is generated

Ω.

η = 1,2,...

there exists

continuous

pseudometrics which

to

symmetric e n t o u r a g e s

a base for

vvtence

by t h e c o l l e c t i o n

runs through a l l

symmetric

c D^

t h e uniform

t h e c o a r s e s t uniform

i s generated

continuous,

of

is

of

i s uniformly

continuous.

uniformly

0^^,02» · · ·

ρ : X χ X -v ]R

a refinement

i s uniformly

In f a c t ures

function

and s o

structures,

Ω

as

asserted. The u n i f o r m : Xj ->• Yj Yj

product i s

functorial

be a J - i n d e x e d

are uniform

spaces.

family

of

in character.

Thus

let

X^

and

f u n c t i o n s , where

Then t h e p r o d u c t

function

Πφ^ : n X j ^ IlYj i s uniformly φ^

continuous

i s uniformly

if

e a c h of

the

functions

Thus t h e p r o d u c t f u n c t i o n

and o n l y i f

each of

the functions

is

a

φ^

is

equivalence.

The p r o o f s e x c e p t for

of

the following

the l a s t w i l l be l e f t

Proposition spaces.

and o n l y i f

continuous.

uniform e q u i v a l e n c e a uniform

if

(2.9).

Let

{Xj}

results to the

(2.10).

bounded uniform

Let

spaces.

{Xj}

ΠΧ^

and

reader.

be a f a m i l y

Then t h e u n i f o r m p r o d u c t

Proposition

are straightforward

is

be a f i n i t e

of s e p a r a t e d

uniform

separated. family

Then t h e u n i f o r m p r o d u c t

of ΠΧ.

totally is

imucea ana comaucea umjorm structures totally

bounded.

Proposition uniformly nXj

is

(2.11).

uniformly

entourage

.

(Cj,rij)

If n.

uniformly

family

for

some £ d"

as

let

and n^ ,

D = JlD^

is

where

the uniform

coinduced

we f i n d

Let

and

Ζ

the

situation

φ : X ·> Y

are uniform

structure coinduced

i s uniformly

continuous

X^

.

for

Thus

X

if

is

rather

less

ψ : Y

spaces. φ

than

Ζ

is

be

functions,

that

from t h a t o f ψφ

to

straightforward.

Suppose

and o n l y i f

without

When we t u r n

the dual c a s e

and

by

structures,

c a n be d e f i n e d

structures

property required in

(2.12) .

of

η = max(nj)

structures,

and s e e k

The c h a r a c t e r i s t i c

basic

η = (η.) a r e p o i n t s of X 3 , since X^ i s u n i f o r m l y

t h a t induced uniform

induced uniform

Y

product

asserted.

therefore,

structures,

X,

non-empty

be a

an e n t o u r a g e

and h a v e t h e e x p e c t e d p r o p e r t i e s .

Condition

of

Then t h e u n i f o r m

D^

)

(ζ,η)

the dual q u e s t i o n ,

ψ

ζ = (ξ

so

We c o n c l u d e ,

where

example, where

connected,

and m u l t i p l e

be a f i n i t e

spaces.

e D^

and

difficulty

for

X = ΠΧ^ , j

connected,

induced

{X^}

connected.

(2.11),

of

each index

is

Let

connected uniform

To p r o v e

then

2i

Y

X.

has

Then

i s uniformly

con-

tinuous . To f u l f i l l "finest"

uniform

Consider, entourages

i n any o f

uniformly

of

this

for

t o g e t any i n s i g h t

of

the uniform

φ

structures

a uniform

At t h i s

level

i n t o the nature

of

Y

i s uniformly

s u b s e t s of

Take f i n i t e

as a b a s e for

satisfied.

which

the family

continuous.

fam i l y is

structure

therefore,

is

(2.12)

t h e c o n d i t i o n one must t r y t o g i v e

Y χ Y

on

Y

the continuous. which are

for which

Φ

intersections

of

members

structure

Y;

then

on

of g e n e r a l i t y the

it

structure.

is

hard

I.M. James For our p u r p o s e s , We s h a l l φ χ φ

however,

the general

of

the entourages

obviously

of

necessary that

Let

R

and l e t

X

D

of

X/R

X

D'

be t h e

there e x i s t s

the images,

form a b a s e

on t h e u n i f o r m

an e n t o u r a g e

for

under

a uniform

π χ IT ,

structure

with this

space

X ,

D'

such

X

if

that for

that

of

on

t h e e n t o u r a g e s of

X/R,

such t h a t

as the q u o t i e n t structure,

X

(2.12)

uniform

as the

quotient,

space.

Examples where a subspace

A

of

(2.13) X

b r e a k s down can be g i v e n by

and s e t t i n g

o b t a i n e d by c o l l a p s i n g = (1,1)

(0,e/2)

,

and

equivalent

,

e

and s o n o t t o

(l,e/2)

Definition X

entourage

is

D

to a point.

D = U,^ .

so that (e/2,1

-

(e/2, e/2)

, so that

Specifically

Then i f U^ ,

choosing

1 -

take

0 < e < i

and

(0,e/2)

s/2)

X/R

both is

belongs

does n o t be l o n g t o

to U, "2

R » U;^ <> R . however,

t o o weak,

(2.14). is

R = Δ υ (ΑχΑ)

belong to ,

whereas

It turns out, (2.13)

A

and t a k e

(l-e/2,1)

to

•> R ο U

space

is

surjective.

of

X/R,

in

exoimple i t

the uniform s t r u c t u r e

s t r u c t u r e and t o

e

for

Y .

L e t us s a y

to this

U

under

natural projection.

We r e f e r

(X,A)

of course;

relation

satisfied.

is

essential.

0 R 0 D' c R ο D ο R .

In t h i s c a s e

uniform

not

form a u n i f o r m s t r u c t u r e on

s h o u l d be

b e an e q u i v a l e n c e

π : X

(2.13)

is

φ

i s weakly compatible w i t h

each

is

o n l y be c o n c e r n e d w i t h s i t u a t i o n s where t h e images

T h i s d o e s n o t happen a u t o m a t i c a l l y ,

R

case

t h a t for most purposes the

and i t

i s usually replaced

The e q u i v a l e n c e r e l a t i o n

c o n p a t i b l e w i t h the uniform of

R ο D' c D » R .

X

there e x i s t s

R

structure

an e n t o u r a g e

D'

condition

by t h e

following.

on t h e

uniform

if

for

each

such

that

nduced and coinduced uniform structures This implies

(2.13),

inverses

the

with

•> R C R = D

is

D'

sufficient

basic a

stated

for

Note t h a t

is equivalent

instead

of

condition

R » D'

by

taking

to the

condition

c D " R.

t o be s a t i s f i e d

when t h e u n i f o r m

structure

Clearly

it

i n the case

of

of

X

i s g i v e n by

base. For e x a m p l e ,

where

Y

and

relation

(φχφ)

uniformly

^ΔΥ

then

π

on is

is

X

If

and X/R

open,

on

X/R

Suppose t h a t entourage entourage

of

X.

D'

R

is

has

of

uniform

with

the

space

is

r e l a t i o n on t h e

the q u o t i e n t

uniform

and u n i f o r m l y

π : X ->· X/R

is

compatible

with the

is

to a given

compatible.

Then

D'

X .

We h a v e

Let

« R c R ο d

(ιιχπ)0'[ΤΓ ( x ) D = π ( ( 0 · " Κ ) [ χ ] )

uniform

uniform

uniform structure

open.

uniformly

structure

the quotient

R

equivalence

compatible

t h e uniform

is

is

b e an e q u i v a l e n c e

with respect

compatible with

structure

R

continuous

projection

φ

projection,

Y.

Let

X .

left

Then t h e

and t h e q u o t i e n t

to

uniformly

the natural uniformly

structure

be t h e

spaces.

d e t e r m i n e d by

(2.15).

space

structure

φ : Y χ Τ -»· Y

are uniform

equivalent

Proposition uniform

let

Τ

product uniform

R

course.

condition

the

entourages,

of

on

Conversely

continuous

and

structure,

then

X

uniform

and t h e

structure. D

for

be a

some

symmeiric

symmetric

= w((R»D)[x])

= w(D[x]) for

all

X c X.

the quotient

Since

uniform

(τΓχπ)Ο'

structure

i s an e n t o u r a g e o f

this

shows t h a t

ττ

is

X/R

in

unifonrily

open. Conversely uniformlv

open,

suppose with

that

respect

π

is

uniformly

t o some u n i f o r m

continuous structure

on

and X/R,

i)

26

I. Μ. James

If

D

is

a symmetric entourage

of

X

then s i n c e

π

is

uniformly

o p e n we h a v e E[m (x) ] c: π Ο [ χ ] for

some s y m m e t r i c

entourage is

of

entourage

X/R

since

an e n t o u r a g e o f

X ,

by u n i f o r m

the

lence that

in

continuity,

the

relations

R

and

S

X/R

and s o

.

D = (wxtr)

and t h e n

(irxiTjD

is

an

On t h e

o t h e r hand i f

^E

an e n t o u r a g e

Ε =

for

forming

is

(πχπ)Ο

following

q u o t i e n t uniform

sense.

on t h e u n i f o r m

are weakly

Let

space

is

compatible, to

relation

.

This

Ε of

completes

by

As we h a v e a l r e a d y section

R

X .

of

the

structure.

(1.13)

seen,

for

entourages

In f a c t

the

For i f

S/R

sane entourage

D'

is

is

X,

equiva-

R c s, is to

. is

so

defined. X,

If,

S/R

is

then

further,

compatible

(2.13).

any u n i f o r m

space

X

the

an e q u i v a l e n c e

compatible with

any e n t o u r a g e

of

be

X/R

X/R

then

and

Ξ

respect

to

X

and

on

constitutes

relation D

to

structures

such that

compatible with

with respect X/R ,

R

X

i s weakly compatible with respect

with respect

for

then

an i n d u c e d e q u i v a l e n c e

S/R

on

of

proof.

transitive

S

Ε

Ε <= (πχπ)Π

X/R

The p r o c e d u r e

If

(χ e X)

of

and s i n c e

X

the

then

R c D'

inter-

relation

uniform D'

and

» D' c D, Δ c

r

we h a v e D'

"RCD"

as r e q u i r e d . uniform X/R

o D ' c D

=

A < ' D c R O D ,

In t h i s c a s e ,

exceptionally,

structure

by t a k i n g

of

the

X

from

induced

D C R O D O R C Q O D O D follows X/R

that

if

R'

is

then the preimage

the

the q u o t i e n t uniform

structure for

the

structure

of

since

every entourage

intersection

(πχπ)

we c a n r e c o v e r

'^R' = R,

of the

D

of

X.

It

the entourages intersection

of

of

Induced and coinduced uniform structures t h e e n t o u r a g e s of Δ'

and s o

X/R

X; is

therefore

separated.

R'

We r e f e r

separated q u o t i e n t uniform

s p a c e of

Proposition

R

(2.16).

on t h e u n i f o r m

Let

space

X .

in the q u o t i e n t uniform equivalence totally

classes

D

Since

X/R

is

of

such

that

X/R =

for

as

the

equivalence

X/R

Also

relation

i s t o t a l l y bounded

suppose that

are t o t a l l y bounded.

all

Then

the

X

is

(R«E)[S] c

bounded s u b s e t X

of

X .

of

X.

Then

symmetric e n t o u r a g e s

E, F

a finite

subset

S

(πχπ)Ρ[π5]

whence X c

of

c ij(E[S]) (D»R)[S]

X ,

such that

Proposition

by

, .

(2.17).

Let

Since

R[S]

(2.3) , there e x i s t s

RLSJ C DCTJ

r e l a t i o n on t h e u n i f o r m formly

Suppose t h a t

t o t a l . l y bounded t h e r e e x i s t s

= 7r((F»R) [ S J )

of

be a c o m p a t i b l e

b e any e n t o u r a g e

F O R - R O E C D O R

Τ

X/R

di-agonal

bounded.

For l e t

X

to

t o the

X.

structure.

RLxJ

reduces

R

,

whence

X ,

a finite

Suppose t h a t

X/R

in t h e c^uotient uniform

structure.

suppose t h a t a l l

the equivalence c l a s s e s

RLxJ a r e

Then

For l e t (πχπ)Ο = Ε

D is

X

i s uniformly

natural projection.

Let

ζ,η

.

is

uni-

Also unifomly

connected.

be a symmetric entourage of an e n t o u r a g e o f

subset

equivalence

connected

connected.

totally

X C (D»D)1;TJ

be a c o m p a t i b l e

space

is a

X/R, e X.

X,

so

that

where

it

denotes

Then

π (ζ )

and

the ττ (η)

be j o i n e d by an E - c h a i n Xj' , . . . So t h e r e e x i s t s

£ X/R . a chain

x^^ , . . .

in

X

such that

ζ ~ χ^ ,

can

28

I. Μ. James

η - x^

and

assert

that

( a , β)

e D,

joined

to

e R " D » R x^

and

for

seme

x^

by a D - c h a i n ,

a D-chain.

So

Although mixing

it

induced

for

i = Ι,.,.,η

be j o i n e d α - x^

by a D - c h a i n .

and

β - Xj^^j^r

and

β

ζ

c a n be j o i n e d t o

is

necessary

- 1.

^nd

α

For

can be

can be j o i n e d t o η

by

by a D - c h a i n ,

to take c a r e ,

and c o i n d u c e d u n i f o r m

as

in general,

structures,

this

i s not

we h a v e i m p o s e d .

Proposition

be a c o m p a t i b l e e q u i v a l e n c e

Let

R

t i o n on t h e u n i f o r m

space

X.

is

saturated

with respect

Then t h e e q u i v a l e n c e with the r e l a t i v e Proposition

to

Let R,

relation

uniform

(2.19).

A

sense that

S = R η (AxA)

on

the

j.

Let

{Xj}

be a f a m i l y

Then t h e r e l a t i o n

uniform product s t r u c t u r e ,

SjRjHj

for each index

The p r o o f s the reader .

of

these

X

relawhich

RLA] C A.

A

is

compatible

structure. of

S u p p o s e t h a t we h a v e a c o m p a t i b l e e q u i v a l e n c e each index

a

Thus we h a v e

be a s u b s e t of

in the

required.

when

problem under t h e r e s t r i c t i o n

(2.18).

I

where

R

on

nXj

(Cj)R(nj)

uniform Rj

on

spaces. Xj

for

i s compatible if

and o n l y

with if

j. two r e s u l t s

a r e e a s y and w i l l be l e f t

to

3.

So f a r , concepts:

The uniform topology

i n t h i s book,

a k n o w l e d g e of

Although i t

is

that

i s not

of

Kuratowski

the unifonn

[13],

a p p r o a c h and t h e n show how t h e u n i f o r m

s e t s of

called

with each subset and c a l l e d closed.

reversing

the c l o s u r e

Η

of

the closure The c l o s u r e

SP

outline

t h e o r y ccmes

X of

on t h e s e t

operator.

a subset H.

Η

When

operator

H, Κ

First

t h e empty s e t

are s u b s e t s of Η υ Κ = Η υ Κ .

By a t o p o l o g i c a l t o p o l o g y on

of

this

in.

2

X

is

of

sub-

X,

containing

of

we s a y t h a t

H,

Η

is

is closed.

two f u r t h e r Secondly

con-

if

then

X

together with a

X.

subset i s closed,

operator

we d e s c r i b e

is

the i d e n t i t y ,

and t h e f u l l

trivial.

so that

every

the topology as d i s c r e t e .

of e v e r y non-empty s u b s e t i s

o n l y t h e empty s e t topology as

operate]

s

s p a c e we mean a s e t

When t h e c l o s u r e

the closure

X

to

associates

atisfy ditions.

the

This

Η = Η

has t o

the

approach

a p p r o a c h a t o p o l o g y on a s e t

by an i d e m p o t e n t o p e r a t o r

X,

theory after

where the c l o s u r e

We b e g i n by g i v i n g

In t h e Kuratowski

topological

prerequisite.

p o i n t of view t h e most n a t u r a l

taken as b a s i c .

determined

a

a c a s e c a n be made o u t f o r

From t h i s

topology i s

topology

i s customary to learn

b a s i c s of t o p o l o g y order.

we h a v e made no u s e o f

set

the

full

are c l o s e d ,

set,

When so

we d e s c r i b e

that the

I.M. James For e x a m p l e , set

Η

when on

of

Η X

X

is

X

define

be an i n f i n i t e

Η = Η

infinite.

in which the

subsets. X

let

when

This d e f i n e s only non-full

For a n o t h e r example,

and l e t

for

all

A g a i n we h a v e a t o p o l o g y on

X.

Η = Η υ Xg

Given a pseudcmetric distance

of

Η

a point

χ

ρ

s e t and f o r

is finite

and

the c o f i n i t e

closed

let

sets

X^

from a s u b s e t

X

Η

Η = X

are the

be a f i x e d

set

sub-

topology

non-empty s u b s e t s

on t h e

each

finite

subset Η

of

of X.

we can d e f i n e

of

X

the

by t h e

formula p(x,H)

= inf ρ(χ,ς) ζεΗ

Then t h e c l o s u r e points

χ

Η

points

X

determines

Η

p o i n t s of

important

is

a closed if

i n o t h e r words t h e H.

X

set.

as

T^^

ρ

is

functions,

Kuratowski

approach.

Definition

(3.1) .

spaces,

subsets

X.

of

Equivalently, of

a closed

φ

is

set

Κ

φ of

Y

is

a l w a y s c o n t i n u o u s when Clearly

the i d e n t i t y

Also the canposition

of

on

function

is

function

continuous

the

are

if

as f o l l o w s

Y,

if

a closed X

the

a metric.

φ : X

continuous

e a c h of

the most important

i s continuous

is

subset

l a s t example

which are d e f i n e d

The f u n c t i o n

are t o p o l o g i c a l Η

if

structure-preserving

in topology but undoubtedly

the continuous

of

Thus a p s e u d c m e t r i c

In t h e

and o n l y i f

t y p e s of

as t h e s u b s e t

X.

a t o p o l o g y on

T^

Several

=0,

from

a t o p o l o g y on

X is

can be d e f i n e d

p(x,H)

at zero distance

We d e s c r i b e

topology

of

such t h a t

of

.

where

φΗ c φΗ

the s e t of

X for

preimage X.

For

are

in

the

and

Y

all

φ ^K example

discrete. i s continuous, functions,

when

X = Y.

when d e f i n e d ,

is

The uniform topology continuous.

Thus we h a v e a c a t e g o r y

topological

spaces

The e q u i v a l e n c e s l e n c e s , or

and t h e m o r p h i s m s

of

the category are

closed

are s t i l l

topological

(respectively

Defin ition

are continuous called

are

functions.

topological

equiva-

homeanorphisms.

A l t e r n a t i v e l y we can c o n s i d e r objects

i n which the objefcts

(3.2) .

are t o p o l o g i c a l

open)

the categories

spaces

spaces,

but the morphisms

functions,

The f u n c t i o n

the

are

in the following

φ : X ->• Y,

i s closed

in which

sense.

where

X

and

Y

where

X

and

Y

if

c ΦΗ for

all

subsets

Defini tion

(3.3) •

are t o p o l o g i c a l

for

all

subsets

In e a c h of becomes e q u a l i t y that

φ

Η

is

of

The f u n c t i o n

spaces,

Κ

if

Y .

these definitions when

φ

the r e l a t i o n

i s assumed

b o t h o p e n and c l o s e d , In c a s e

Φ

or c l o s e d ,

open,

φ ; X ·+ Y,

i s open

of

discrete. is

X .

φ

in

all cases,

when

of c o u r s e ,

to say t h a t

Thus a c o n t i n u o u s o p e n or c l o s e d

inclusion

t o be c o n t i n u o u s .

is bijective, is

of

φ ^

bijection

is is

Note Y

t o say

is that

continuous. a

topological

is

a set

equivalence. Let is

φ : X ->• Y

a topological

coarsest

be a f u n c t i o n ,

space.

The i n d u c e d

t o p o l o g y for which

that the closure Η = φ"^ (φΗ) t h e o r e i m a a e of

Η

of

where

φ

is

a subset

X

t o p o l o g y on

continuous. Η

of

X

X

and is

Υ

the

T h i s means

i s given

by

, the closure

of

t h e image i n

Y.

For

example,

32 if

I . James Y

is

a point-space

topology. istic

Proposition

tinuous

and

of if

analogous to

(3.4).

X, Y

topology

X

important

Let Ζ

is

is

in

of

induced

is

is

character-

ψ : Y ->• Ζ

spaces.

that

of

is

be

functions,

Suppose t h a t

Y.

Then

injective

is

terminology. if

ψφ

is

the con-*

sufficiently

We s a y φ

is

that

injective

and

topology. subsets

of

topology,

space

topological

Specifically, X

spaces are

usually called

the

if

A

then the c l o s u r e

the intersection with

A

of

always

relative is

in

a subset

A

of a

the closure

of

subset of

Η

X

is

X. L e t u s now r e t u r n t o t h e u n i f o r m

uniform

space

is defined

a t o p o l o g y on

by t a k i n g

each subset D

trivial

continuous.

φ

special

in t h i s case.

A

and

a t o p o l o g i c a l embedding

the t o p o l o g i c a l Η

ψ

c a s e when

In p a r t i c u l a r , given the

Y

i n d u c e d from

has the induced

topology

the

).

are t o p o l o g i c a l

to deserve

φ : X ->- Y

(2.1

φ : X

and o n l y i f

The s p e c i a l

X

topology is

The i n d u c e d t o p o l o g y h a s t h e f o l l o w i n g

property,

where

the induced

Η

of

X,

called

the c l o s u r e X,

i s the

where

R = nD

but g e n e r a l l y

uniform

t h e uniform structure If metric

structure

is

topology

the t r i v i a l

where

Η

contains

entourages,

with

topology,

with the t r i v i a l

the

while

uniform

topology.

s t r u c t u r e on

t h e n t h e t o p o l o g y on

w i t h t h e uniform

the

associated

the d i s c r e t e

topology associated

is

of

for

subset.

the uniform

the uniform ρ

the i n t e r s e c t i o n

as a proper

For e x a m p l e , discrete

is

H,

η DCH],

Of c o u r s e

a

topology,

i n which

intersection X.

If

t h e unifor m

operator

runs through t h e e n t o u r a g e s of

R[H],

category.

X X

i s defined

by a p s e u d o -

d e t e r m i n e d by

p

toDoloav a s s o c i a t e d w i t h the uniform

coincides structure

The uniform topology d e t e r m i n e d by

ρ.

Topologically

Different t o the line

discrete

uniform

but not uniformly

structures

same t o p o l o g y . E,

33

of

w h i c h i s g e n e r a t e d by t h e f a m i l y ΔΕ υ ((α,<») χ where is

α

on t h e same s e t may g i v e

For e x a m p l e ,

with the coarsening

take

associate

the

t o be t h e

of

uniform

rise

real

structure

subsets

(α,") ) ,

the d i s c r e t e

be an i n f i n i t e

X

the d i s c r e t e

r u n s through t h e r e a l numbers;

still

discrete.

topology.

set.

the uniform

For a n o t h e r

With each p a r t i t i o n

example

{η^}

of

X

topology let

X

we can

subset

u(H. X H.) j of

XXX.

For u n r e s t r i c t e d

partitions

the d i s c r e t e uniform

structure.

generate

uniform

uniform

a different

topology remains

Proposition structure uniform

is

(3.5).

Let

separated

topology.

if

these subsets

For f i n i t e

structure,

partitions

but the

generate they

associated

discrete. X

be a u n i f o r m

and o n l y i f

X

space.

The uniform

is

in

Tj^ ,

the

I.M. James For < = R[x] ic ,

if

if

R

is

the i n t e r s e c t i o n

for

each point

and o n l y i f

much t o reccmraend i t .

sets

e X .

R = Δ ,

As an i n t r o d u c t i o n

where i t

χ

to

the entourages

Since

R[x] = χ

the r e s u l t

follows

at

for

approach

and o p e n s e t s

are t h e i r complements,

axians follow

a t o n c e from

t h e p r o p e r t i e s of

eind open

closed

of c o u r s e .

The

the

has

stage

to closed

We h a v e a l r e a d y d e f i n e d

all

once.

we h a v e r e a c h e d t h e

t o be a b l e t o r e f e r

i n t h e u s u a l way.

then

,

t o p o l o g y t h e Kuratowski

However,

i s convenient

of

sets

usual

closure

operator. In p a r t i c u l a r , of

X

is

of

X,

that

is

formed

is

fundamental.

χ

(3.6).

neighbourhood

Let

filter

subsets

,

D[x],

We h a v e

Then

Η

if Η .

in

t h e uniform D

X

just

if

the

space.

every

subset

runs through the

is

of

result

Then

topology,

filter

We s a y

The f o l l o w i n g

be a u n i f o r m

where

of

is

X

the complement

χ < Η ,

o p e n and c o n t a i n e d neighbourhood of hence

χ

such t h a t c υ , DCX],

ϊί .

since

χ /

in

DLX] .

χ .

X ^ D[H] .

Then

for

anv s u b s e t

Η

of

X .

D[x]

and s o

Conversely,

Η

the

generated

entourages

DCH]

,

with

D

let

U

is closed

does not meet

D ,

symmetric.

χ £ X - Η , which

and

DCX]

We r e f e r

as a uniform

as a uniform

is

be an o p e n χ < Η ,

a symmetric entourage

the proof.

any e n t o u r a g e to

of

Η = X - U

Then

which completes

Η

DLH] ,

So t h e r e e x i s t s

More g e n e r a l l y we r e f e r for

H.

a subset

x.

χ

X. For c o n s i d e r

hood

meets

the neighbourhood

which c o n t a i n

by t h e a d h e r e n c e p o i n t s o f

Proposition

D[x]

N^ ,

by t h e o p e n s e t s

of

X

of

has a t o p o l o g y then each p o i n t

an a d h e r e n c e p o i n t o f

neighbourhood

by t h e

X

equipped with a f i l t e r

generated X

if

D and

to the

of

X

so neighbour-

neighbourhood.

neighbourhood of

Η

The uniform topology A topological neighbourhoods

Proposition is

χ

X

is

said

generate

(3.7).

Let

X

t o be r e g u l a r

N^ ,

for

each

be a uniform

if

the

point

χ

space.

closed of

Then

X ,

X

regular. For

X £ X, Ε

cf

space

let

D[x]

where

D

such that

hood of

X

be is

a uniform

neighbourhood

an e n t o u r a g e o f

E^ c D .

Then

such t h a t

X.

ELXJ

Ci.(E[x])

is

of

the

Choose

an

a uniform

c D[x],

which

point entourage

neighbour-

establishes

regularity. It is

follows

separated

fran

if

the

last

and o n l y

if

three

it

is

results

that

Hausdorff,

a uniform

in

the

space

uniform

topology. Topological if

{Xj}

is

and the

of

topological

filter

of

the point

Uj = X j

Hj c X j of

nUj,

c a s e of

indices

where for

each

j

,

the

j

The u n i f o r m to products. spaces

all

a subset

for

are defined

a feunily

neighbourhood products

products

Uj

is

but

ΠΧ^

and

of

closure

of

topological

Proposition of

(3.8).

the uniform

space

ΠΧ^

Let X.

all

in

of

is

Πχ^

indices

j.

HH^,

where

but

a finite

a family with

the X^ In

number

ΠΙΙ^ = IIIK in

relation

of

uniform

t h e uniform

coincides with the product

Η

the

by in

satisfactorily

{X^}

Thus

x^

i s g i v e n by

topology associated

on

the associated

for

way.

i s generated

form

operator

if

product structure

then

number o f

the

behaves

Specifically

then the uniform

(Xj)

H^ = X^

topology

spaces

usual

a neighbourhood

a finite

of

in the

topology

spaces.

be a u n i f o r m l y

Tlien t h e c l o s u r e

connected Η

of

subspace

11

is

also

u η i f ormly c onnec t e d . For

let

λ

: ii ->· D

be u n i f o n n l y

continuous,

where

X)

is

.

36

. James

discrete. constant

Then

λ|Η

H,

since

on

D = (λχλ)

is

i s uniformly continuous, Η

(χ,ξ)

£ D

λ

c o n s t a n t on

is

Proposition

some H,

connected uniform

To s e e the Xj

all

for

then

and s o

λ(χ)

= λ(ζ).

which proves

(3.8)

(X^)

choose a point X

consisting

a p r o d u c t of

of

number o f v a l u e s o f

For e a c h f i n i t e set

J

Xj = ζ^

equivalent

to

subset

by

by

(3.8).

seen,

proves

X = ΠΧ^

in the a s s o c i a t e d Η

and

χ = (x^)

j

such

j

.

I Uj

so

nUj

that

If

U^ = Xj

then

and

consider

for for

at

meets

Η .

Η

consisting

X. Now

Η

of

points

is χ X.

Η

the union of

is

the point Η

i s dense in

(Xj)

and s o

ξ. is

the

uniformly

* ...

( 2 . 4 ) , and s o

Since

of

j

them c o n t a i n s by

(η = 1 , 2 , . . . )

is these

Therefore

Η

uniformly

X ,

a s we h a v e

be u n i f o r m l y

continuous,

(3.9).

(3.10). Y

X

such t h a t

, · ••.j

(2.11).

connected,

connected,

For l e t

j

the uniform product

and e a c h o f

and

(j

whenever

connected,

uniformly

X

Therefore

X .

the s u b s e t of

such t h a t

where

so

non-empty uniforml;

of

points

open s e t s

i s dense in

Η

Proposition

a

χ e D[H],

number of v a l u e s o f

Therefore

this

of

ζ = (ξ^)

number o f v a l u e s of

is

is

.

be a f a m i l y

most a f i n i t e

subsets,

Δ

Then t h e u n i f o r m p r o d u c t

a t most a f i n i t e

is

uniformly

is

Now

since

χ « Η

Let

of

but a f i n i t e

indexing

H,

λ

connected.

Η

nUj c nXj

So i f

spaces.

this,

subset

D.

ξ £ H,

(3.9).

is uniformly

connected.

a s y m m e t r i c e n t o u r a g e of

s y m m e t r i c e n t o u r a g e of for

i s uniformly

and s o

Let

φ : X -·• Y

are uniform uniform

spaces.

Then

φ

is

continuous,

topologies.

be a s u b s e t

of

X.

If

Ε

i s any e n t o u r a g e

The uniform topology of

Y

then

D = (φ^φ)

^E

i s an e n t o u r a g e o f

X

such

that

false,

as

φΟ[Η] c Ε[φΗ] Therefore

φΗ c φΗ,

However,

as

t h e c o n v e r s e of

every undergraduate X = Y =JR,

required.

student

the r e a l

f u n c t i o n g i v e n by

line,

φ(t)

uniformly

continuous,

Similarly

for

therefore,

functions.

Under t h e

topological

equivalences.

s p a c e s and u n i f o r m l y

of

and

Y

(3.11).

Let D[xJ

φ '•R

i s uniformly

that φ(x)

Let

R .

i s defined

from

continuous and

continuous

equivalences

become

uniform

t o t h e c a t e g o r y of

topolog-

a s shown by

φ : X

Y

be u n i f o n n l y

Then

where for

Κ c γ

φ

and

each entourage

open t h e r e e x i s t s

and s o

I χ Μ

frcra t h e c a t e g o r y of

spaces.

Ε[φ(x)] C φΟ[χ]. e K,

but not

structure.

spaces

uniform

squaring

is

open,

open,

in

where

the

topologies.

X e φ '•K,

meets

i s continuous

function

open f u n c t i o n s

are uniform

a s s o c i a t e d uniform

φ

topological

s p a c e s and o p e n f u n c t i o n s ,

Proposition

t o be t h e

and u n i f o r m l y

functor

take

φ

that a functor

functor,

There i s a s i m i l a r

φΟ[χ]

Now meets

χ e X. D

of

I assert X.

an e n t o u r a g e

Ε[φ(χ)] K,

meets

hence

Ε Κ

that

For

since

of

Y

such

since

D[x]

meets

asserted. In p a r t i c u l a r a s u b s e t

if

then

spaces

to the category

as

and t a k e

For e x a m p l e

in the Euclidean uniform

functions

φ

taught.

= t^;

t h e c a t e g o r y of u n i f o r m

X

is

the m u l t i p l i c a t i o n

We c o n c l u d e ,

ical

(3.10) i s g e n e r a l l y

it

i s uniformly

aenerallv

true,

open.

A

of a uniform

However,

a s c a n be s e e n from

space

the converse (1.16).

In

X

is

is

open

not

fact,

if

A

38

.Μ. James

i s uniformly

open i n

in

compact spaces t h i s c o n d i t i o n

as

X ;

for

X

A

and

X - A is

are

open

sufficient

as

well

necessary. A function is

usually

local

further

each p o i n t

φ.|υ

injective.

is

said

φ : X

spaces,

to

χ

e X It

uniform

spaces,

respect

to the uniform

then

D.

If

X

to uniformly

i s uniformly

For example

the

is a local

topological or more

U

such

and

(3.11)

where

X

topological

and that

that

and

if

Y

are

equivalence,

with

topologies. X

is

space

said

t o be c o n n e c t e d i f

is constant,

a uniform

connected.

space

this

functions

for

applies

However, n o t e v e r y

line

structure,

φ

each in

partic-

but not c o n n e c t e d

that

uniformly

i n the uniform

i s uniformly

every

discrete

and s o we i n f e r

is connected,

rational

Euclidean uniform

are

equivalence,

(3.10)

equivalence,

continuous

connected uniform

Y

i s c o n t i n u o u s and o p e n ,

from

λ : X -<· D is

and

admits a neighbourhood

space

continuous function

φ

follows

φ

X

topological

if

i s a l o c a l uniform

The t o p o l o g i c a l

ular

where

homeomorphism,

Y

space

Y ,

be a l o c a l

φ : X

X

then both

topology.

connected,

in

the

i n the uniform

topology. Most o f topological (2.7)

the r e s u l t s theory.

proved e a r l i e r

For e x a m p l e

the t o p o l o g i c a l

in

the

analogue

of

is

Proposition with

X

(3.12) .

Let

X, Y

connected.

Let

α, 0 : X

and l e t

φ : Y

φα = φβ

then the coincidence

either

empty or

Definition for

have a n a l o g u e s

Ζ

be a l o c a l

and

Ζ Y

be t o p o l o g i c a l be c o n t i n u o u s

topological set

Μ(a,β)

spaces,

functions

equivalence. of

α

and

If β

is

full.

(3.13).

each entourage

The u n i f o r m D

of

X

space

X

there e x i s t s

has property a finite

S

if

covering

The uniform topology of

X

by c o n n e c t e d D - s m a l l

The

S

i e n t for ages,

here

s t a n d s for

the s t a t e d

a unifonn

example, family

it

is

structure

sufficient

Euclidean uniform

of

X

for

as

Definition

(3.14).

line

and s i m i l a r l y

In t h e u n i f o r m

The u n i f o r m

there exists

space

X

Proposition

(3.15).

then

locally

DCx]

DCX]

a base for

contains

be a s y m m e t r i c

X

has property

for

all

indices

neighbourhood of Κ C

in

the

topological

invariants, localize

S

DCx]

space

such t h a t

subsets.

each

X

has property

of

χ £ X.

such

Now

Η . ^

H^ Κ

χ ε H^ .

that <= D C X ]

χ

I .

S

covering is

assert lor

Ε « Ε •> ε c D .

a finite

Let

such t h a t

(3.15).

such

D.

a c o n n e c t e d neighbourhood of

H^ c ECH^] .

χ

locally

structure

i s connected for

neighbourhood

there e x i s t s

j

i s unifonnly

connected.

u_ Η . <= u_ xeHj ^ xeH^

This proves

S,

dimensions.

t h e uniform

the uniform

entourage

by c o n n e c t e d E - s m a l l since

If

be a u n i f o r m

Ε

D-small,

e.

in higher

X

and e a c h b a s i c e n t o u r a g e

that

finite

in

of

Let

for

t h e o r y we n e e d t o

point

is

entourIn t h e

have property

neighbourhood

X

basic

each p o s i t i v e

t h a t the uniform χ

suffic-

by a p s e u d o m e t r i c ,

a s t a n d a r d way t o l o c a l i z e

uniformly,

is

can be c o v e r e d by a

the r e a l

structure,

in appropriate c a s e s .

connected i f

if

for

it

i s g i v e n by a b a s e .

open e - b a l l s ,

Thus bounded i n t e r v a l s

Clearly

t o be s a t i s f i e d

structure determined

of c o n n e c t e d

There i s

Sierpinski.

condition

when t h e u n i f o r m

c a s e of

subsets.

let

Since

{H^}

of

X

E - E - E - s m a l l and

so

be the union of Then

Κ

is

the

H^

a connected

4U

I. Μ. James Clearly

if

X

has property

In t h e o t h e r d i r e c t i o n Proposition If

X

For

let

entourage

D

X.

Ε

locally

X

{H^}

of

Ε ο Ε c D . contains

is

j

is

covering

a finite

FtXj]

is

Ε ο Ε c D,

and s o

Proposition

(3.17).

the subset logical

To s e e if

(ζ .η)

X

this,

is

topological

x^ c tK

is

Let

X

D

of

by

(3.17),

be a u n i f o r m

X

i s a neighbourhood

from t h e f i r s t

adherence p o i n t of

Μ

(ζ,η)

some

and t h e n

finite For

Moreover and

space.

of

For

Μ Μ

of

each X χ X

in the of

t D · Μ · D

(a,β)

X

E^

is

£ Μ.

if

topo-

such

(a,β)

in

only

the It

form a b a s e f o r a symmetric

follows the

uniform

entourage

a neighbourhood

and t h e o b s e r v a t i o n t h a t and o n l y i f

and

assertion.

The s e c o n d

Μ .

Since

In p a r t i c u l a r each

the diagonal.

if

χ

[F[x.J}

F c Ε

and e a c h s u b s e t

Ε c Int D . of

a

c o i n c i d e s w i t h t h e c l o s u r e of

for

that

obtained.

observe that

E^ c D

such

each point

Moreover t h e i n t e r s e c t i o n D

locally

F

subsets.

a product neighbourhood of

and s o

uniformly

Then

i s a neighbourhood

all

S.

be an

subsets.

since

s.'^nje g i v e n D t h e r e e x i s t s

such t h a t

follows

.

square t h i s proves the f i r s t

Ε

is

Ε

connected for

a t o n c e t h a t t h e open e n t o u r a g e s of structure,

X

space.

has property

and l e t

by c o n n e c t e d

(3.16)

c DCa] x DCg]

D [ a ] X DCS]

is

and s o D - s m a l l ,

X χ X.

neighbourhoods for

X,

X

by n o n - e m p t y F - s m a l l

D · Μ ο d

square

bounded.

bounded u n i f o r m

then

If

F[x]

of

symmetric entourage

totally

a symmetric entourage

choose a point

F-F-small

is

t o t a l l y bounded t h e r e e x i s t s

X

each index

X

be a t o t a l l y

connected

neighbourhood

Since

covering

X

b e any e n t o u r a g e of

connected then

of

Let

such t h a t

t h e uniform

then

we h a v e

(3.16).

i s uniformly

S

of

Ε ,

entourage

assertion (ζ,η)

0[ζ] x 0[η]

is

meets

an Μ

The uniform topology for

all

synunetric e n t o u r a g e s

entourages

form a b a s e f o r

Proposition

(3.18).

space

X ,

i n the uniform

X X X

of

structure, For So l e t

form

Ε

A

of

A ,

the c l o s u r e

of

Ε

frctn w h i c h

(3.18)

of

follows

since

A

A .

Then

Ε

at

is

function,

less

X

of φ ^H

Y

is is

open open

Y

(resp. (resp.

take

closed) closed)

open

(resp.

with

closed)

the saturated necessarily

open

ir

of

(resp.

uniform

i f and o n l y in

X.

topology However

Y

on

is generally if

i s unifonnly

R

X

closed)

two t o p o l o g i e s

on

Υ

X ,

quotient

be a

its

By thi

(2.1i). relation

In R

projection.

s e t s of

Η

preimage

This has the

a uniform

X.

expected

particular, on

X

and

Then t h e π

of

Note t h a t

φ : X •>• Y

coarser

s p a c e and

X .

i s defined

Y

is

an

coincide.

R

is

a weakly

Then t h e

but the a s s o c i a t e d

than the q u o t i e n t

and s o o p e n ; X/R

Then

function. is

X/R

if

i s c o m p a t i b l e w i t h t h e uniform

open,

X.

are t h e images under

e q u i v a l e n c e r e l a t i o n on

structure

of

φ : X

has the q u o t i e n t t o p o l o g y i f

open or c l o s e d c o n t i n u o u s

compatible

D

s p a c e and Y i s a s e t .

the natural

sets

Now s u p p o s e t h a t

X .

we mean t h e t o p o l o g y i n w h i c h a s u b s e t

t h a t we h a v e an e q u i v a l e n c e

Y = X/R,

in

contains

t o p o l o g y and t h e

Let

c h a r a c t e r i s t i c property analogous to suppose

i s dense

once.

is a topological

q u o t i e n t t o p o l o g y on

uniform

and s o i s an e n t o u r a g e o f

straightforward.

where

in

X .

X χ X

D

the uniform

in the r e l a t i v e

an o p e n e n t o u r a g e

contains

well.

Then t h e c l o s u r e s

The r e l a t i o n b e t w e e n t h e u n i f o r m topology

s u b s e t of

t h e e n t o u r a g e s of

i s dense in

A χ A

t h a t the. c l o s e d

s t r u c t u r e as

be a d e n s e

be an e n t o u r a g e of

t h e t r a c e on

follows

topology.

a base for

A X A

It

the uniform

Let

the entourages

D .

quotient uniform

«.opology.

structure

in this case therefore

then the

π

42

I. Μ. James For excunple, c o n s i d e r = (0,")

uniform for

is

has the m u l t i p l i c a t i v e

structure.

all

λ £

trivial,

Hausdorff

.

of

that on

if

X

only

if

n o t i o n of

p o i n t of

F

if

for

all

X

is

for

p o i n t of

G

p o i n t of

F .

F

(3.19).

space

t h a t e a c h of

X.

contained

χ in

of we

say

i s conpact

F

We s a y t h a t filter

H^

converges

Let

χ

i s most Γ

and

limit by

F,

Clearly

p o i n t of G

if

confused

is a

x.

in

The

i s refined

to

F

F,

s h o u l d n o t be

is

if

F

is

an

adherence

of

a

limit

canpact-

convenient.

be an o p e n c o v e r i n g

Then t h e r e e x i s t s

an e n t o u r a g e

the uniform neighbourhoods

i n some member of

i n terms

i n which the d e f i n i t i o n

n e s s through open c o v e r i n g s

Given

X

w h e r e a s any a d h e r e n c e p o i n t o f

Proposition

convenient

e a c h member o f

t h e n any l i m i t

We b e g i n w i t h a r e s u l t

uniform

of

filters,

limit point.

of

although

Specifically

HE F. Then

the neighbourhood

a refinement

is

ccmpact-

a d m i t s an a d h e r e n c e p o i n t .

and i n t h a t c a s e we s a y t h a t G

details).

an a d h e r e n c e p o i n t o f a f i l t e r

adherence p o i n t ,

w i t h the n o t i o n of

structure

the

i n the Appendix,

others.

an a d h e r e n c e p o i n t

on

all

definition

is

every f i l t e r

x Ε

(λξ,λη),

satisfies

open c o v e r i n g s

of

χ £ Μ

is

r e l a t i o n on

r e s u l t s which i n v o l v e

χ

o t h e r words i f

X

for

additive

the points

5 for

the a l t e r n a t i v e

i s more c o n v e n i e n t

a point X

i n terms of

where

t h e q u o t i e n t uniform

As e x p l a i n e d

the usual d e f i n i t i o n

x E,

has the

topology

c o n d i t i o n . (See S e c t i o n

in sane f a s h i o n .

filters

Κ

are

In t h i s c a s e

whereas the quotient

f o r many p u r p o s e s ,

and

(ζ,η)

We c o n c l u d e w i t h a s e r i e s o f ness

product

Use t h e e q u i v a l e n c e

which the r e l a t i v e s

for

the uniform

D[x](x£X)

of

the

conpact

D

such

is

contained

Γ.

there e x i s t s s a n e member o f

a uniform Γ.

Let

neighbourhood Ε

^^jjtx] of

be an e n t o u r a a e

χ

such

The uniform topology that

Ε » Ε X X

covers by is

c D X

t h e compact

x,,...,x

,

special

X.

Extract

say.

(3.19)

case will

Proposition and

Y

Then t h e

of

a finite

be f a m i l i a r

(3.20).

Let

Ε

entourage

of

Y

with

of

entourage

such t h a t

.

D

of

X

t φ~^Ρ[φ(χ)],

φ(X)

e Γ[φ (ς ) ] ,

χ

this

is

X

and s o ,

izable

structure

that

if

the uniform

fashion.

as

φ

Since φ i s

is

in

V^

φ(ζ)

continuous

neighbourhood

constitutes

by ( 3 . 1 9 ) ,

an open

there e x i s t s

an

neighbourhood for

seme

e Ρ[φ(χ)]

x. and

Thus so

.

such

i s uniformizab le

that

the uniform

discrete

a compact t o p o l o g i c a l

in order

Before

i s unique.

if

and

it

topology

and t r i v i a l

a s we h a v e s e e n .

structure

{ φ χ φ ) 0 c E,

asserted.

space

For e x a m p l e ,

s e e compact r e g u l a r

natural

X

be a symmetric

we o b t a i n t h a t

a topological

s p a c e m u s t be r e g u l a r , shall

ζ

are u n i f o r m i z a b l e ,

(3.20)

then

all

continuous,

topology.

spaces

o n c e from

a

therefore

be g i v e n a u n i f o r m

ical

F

: χ e X}

X

in p a r t i c u l a r

i s uniformly

the g i v e n

which

Then

X , an o p e n

fv^

i s contained

true for

We s a y t h a t

and l e t

of

φ Ο [ ζ ] c F » Ε[φ ( ζ ) J <= Ε[φ ( ζ ) ]

φ

''η

of

compact.

such t h a t e a c h uniform

ζ e X,

0[ς]

so

η Ε

be c o n t i n u o u s w h e r e

X

Y,

The f a m i l y

the conpact

where

Since

η ...

analysis.

F » F c E.

for each p o i n t

c φ~^Ε[φΙχ)]

0[ζ.|,

Ε ''l

indexed

property.

frctn r e a l

b e any e n t o u r a g e o f

exists,

covering

subcovering

intersection

φ : X -<· Y

spaces,

Ε„Εχ](χεΧ) χ

continuous.

Let

there

neighbourhoods

t o p r o v e an i m p o r t a n t r e s u l t

a r e uniform

uniformly

X

The f a m i l y

an e n t o u r a g e w i t h t h e r e q u i r e d We u s e

V

.

It

t h i s we p r o v e

the

and a s we

spaces are always uniformizable establishing

at

uniform-

Of c o u r s e

t o be u n i f o r m i z a b l e ,

is

topolog-

follows

space i s

can

in a

another

Ht

t.Μ. James

result,

which demonstrates

n e s s i n t e r m s of

adherence

Although uniform general

normal:

uniform exists

spaces

closed

we c a n

(3.21).

space

are r e g u l a r

X,

an e n t o u r a g e

D

ccmpact-

they are n o t

in disjoint

the s u b s e t s i s

compact

prove

Let with

of

s u b s e t s do not admit

However when o n e o f

than c l o s e d ,

Proposition

the d e f i n i t i o n

points.

disjoint

neighbourhoods. rather

t h e u s e of

Η, Κ Η

be d i s j o i n t

closed

of

X

and

Κ

such that

s u b s p a c e s of

compact. D[H]

the

Then

and

DLKI

there are

disjoint. For

suppose n o t .

symmetric entourage

Then D

of

s y m m e t r i c e n t o u r a g e s of a filter

F

on

X

t h e uniform

sects

11

and s o

Corollary space of

X,

as

Proposition

So a s

D

t r a c e of

Then f o r

since

Η

Let

(3.23).

Κ

diction,

is closed.

as

Let

X of

of

Μ

generates

filter

has

entourage χ

inter-

T h u s we h a v e a

D

the

runs through the of

the diagonal

Δ

uniform entourages

K.

in

space.

X χ X

Then consti-

X.

of

i t does not

generate

Κ

of

be a ccmpact r e g u l a r

(1.1).

satisfy

symmetric open neighbourhood Ν of (MoM)\N

the

on

this

D ° D ° D[x]

the neighbourhoods

two c o n d i t i o n s that

D •> DLH]

each symmetric

Obviously the neighbourhood f i l t e r first

each

runs through

be a c o m p a c t s u b s p a c e o f

D[K],

filter

tutes auniformization

the

for

required.

a base for

the neighbourhood

D » D[H]

neighbourhood

The s u b s e t s

form

the

meets

By c o n p a c t n e s s

χ e H,

(3.22).

X.

X

χ e K.

of

contradiction,

X.

Κ .

an a d h e r e n c e p o i n t D

Κ

a filter

F

r u n s t h r o u g h t h e members o f

W

of

Suppose, the t h i r d .

Δ

to obtain a

contra-

Then t h e r e e x i s t s

Δ sueh t h a t the

on t h e c o m p l e m e n t Ν .

satisfies

intersections CN

By c o m p a c t n e s s

of there

N,

as

a

The umjorm topology exists Then

an a d h e r e n c e p o i n t ς /

{η)

since

(ξ,η)

η /

ΝΕξ]

Hence t h e r e e x i s t d i s j o i n t respectively. bourhoods L = X -

Since

H, Κ

(HuK)

Δ.

is

ς,η

,

F ,

and

regular

υ (VxV)

υ

η Μ = Η X U,

and s o t h e n e i g h b o u r h o o d Μ' = Μ » Μ . us o u r

ζ,η,

closed .

neighWrite

neighbourhood

(LxL)

Since

(ΧχΚ)

η Μ = V x Κ

Η χ Κ

Μ' η CN

of is

(ξ,η)

d o e s n o t meet

a member o f

F

this

gives

contradiction.

Clearly

the uniform

topology since

X

for

verse,

let

topology. X

of

Κ c ν

N e i g h b o u r h o o d o f d i a g o n a l b u t n o t an

of

similarly.

U, V

Η c u,

«· X .

Now

(HxX)

given

ζ,η

there e x i s t

such that the

where

η / ΤξΤ

neighbourhoods

and c o n s i d e r

Μ = (UxU) of

of

X

of

if

t o p o l o g y c a n n o t be f i n e r χ e X

each neighbourhood V

then Ν

of

i n t h e same t o p o l o g y ,

V

contains

than

N[x]

is

Δ .

To p r o v e

b e an o p e n n e i g h b o u r h o o d o f

By r e g u l a r i t y

entourage.

χ

a neighbourhood the

in the

the c l o s u r e

and s o t h e c o m p l e m e n t

the

Ν

given

{x} of

con-

of

I.M. James {χ}

χ CV

this

is

shows

a neighbourhood

t h a t the g i v e n

of

topology

t o p o l o g y and s o t h e t o p o l o g i e s By way o f Proposition space.

X

is

locally connected, For

let

D

Let

X

locally

connected

of

uniform

required.

then

F^^CxJ

X,

for

sets containing

and s o i s χ

the

If

uniformly

X

a connected

each point

Ε

be a

is

locally

uniform

χ o f X.

Now t h e

,

r u n s t h r o u g h t h e p o i n t s of

the diagonal

is

and l e t

Ε = Ε c D.

by d e f i n i t i o n , E[xJ

X

topological

uniformization.

be any e n t o u r a g e o f

Ν = υ (F [ x ] X F [ x . l ) Λ A χ

as

c ν

than the

be a c o n p a c t r e g u l a r

in the unique

there exists,

neighbourhood

where

N[x]

i s no f i n e r

coincide

symmetric entourage such t h a t connected

Since

a p p l i c a t i o n we p r o v e

(3.24).

If

Δ .

X,

is

an e n t o u r a g e .

a neighbourhood

As u n i o n s o f

connected

vmiform n e i g h b o u r h o o d s N [ x J a r e c o n n e c t e d .

Since Ν c uiECx] X E [ x ] ) t h i s proves If

if

(3.24) .

we c o m p a r e

conclusion and o n l y

c Ε ° Ε c D

(3.24) with (3.15)

t h a t compact r e g u l a r if

might wish t o

S .

local conditions

b u t h e r e we m e n t i o n j u s t

(3.25).

ccmpact

if

there e x i s t s

uniform

neighbourhood

its

connected be m e e t i n g

n e a r t h e end of in c a s e

the

Section the

8,

reader

properties.

The u n i f o r m

space

an e n t o u r a g e D[x]

we r e a c h

We s h a l l

one more e x a m p l e ,

investigate

Definition

(3.16)

spaces are l o c a l l y

t h e y have p r o p e r t y

some o t h e r u n i f o r m

and

X D

of

i s compact for

i s uniformly X

locally

such t h a t

each p o i n t

χ

the of

The uniform topology Any t o p o l o g i c a l namely t h e c o a r s e s t function

α : X

Κ

on

intersections

X X X ,

topology,

jR .

of

such t h a t

ο

χ

: X

t X

Κ

- α(χ)|

the

is

the

g e n e r a t e d by

form

- α(η)|

< c}

D [x] α, €

is

e

is

positive.

open i n the

uniformizable topological

In g e n e r a l , of

the

given

space

X

set

is Η

of

throughout

the

£,α(χ)

+ t)

as c o a r s e as latter

a topological

the

is a

X

We r e c a l l

and e a c h p o i n t

real-valued Η

s p a c e t o be that a

t o be c o m p l e t e l y r e g u l ; u r

and

χ

function

α = 0

spaces are obviously regular,

g i v e n of r e g u l a r

least

however,

regularity. said

there e x i s t s a continuous α = 1

is at

-

former.

i s complete

each c l o s e d

< e} = α " ^ ( ο ( χ )

topology

The p r e c i s e c o n d i t i o n f o r

regular

with

i s c o n t i n u o u s and

the subset

uniform

topology.

proper refinement

that

continuous

continuous,

of

structure,

since

so the associated

for

each

This structure

subsets

D^ [χ] = 1ς:|α(ς) ut , C

original

be g i v e n u n i f o r m

e X X X : |α(ς)

where

For e a c h p o i n t

can

i s uniformly

" ί(ζ,η) of

X

structure

Euclidean structure finite

space

at

χ

if

of

X - Η

α : X

Κ

.

such

Completely

w h i l e e x a m p l e s c a n be

s p a c e s which are not c o m p l e t e l y

regular.

We p r o v e Proposition if

(3.26).

and o n l y i f For s u p p o s e

uniform

is completely

that

X

For l e t χ

Η

be a p o i n t

space

X

J.s

a b o v e i s no c o a r s e r be a c l o s e d

of

uniformizable

regular.

is completely regular.

topology defined

topology. and l e t

X

The t o p o l o g i c a l

X - Η .

Then t h e

than the

set in the given Since

X

is

given topology

completely

to

i.tvi.jumts

regular that

there e x i s t s

α =

1

a continuous

throughout

Η

function

and

α = 0

α : X

at

χ

®

.

such

Consider

the

entourage

D = If

ς £ D[x]

and h e n c e

= ίΙζ'Π)

e X X X :

we h a v e

(χ,ί)

|α(ζ)1

coincides

<

.

ε D

It

with the given

1α(ζ)

- ο(η)1

so t h a t

follows

<

U

|a(x)

- α{ς)|<

t h a t t h e uniform

topology.

This proves

^

topology

(3.26)

in

one

direction. In t h e space,

and u s e a v a r i a n t

Urysohn, uniform D[x] and

other direction

as

follows.

topology,

c X - Η , then,

of

X

Each p o s i t i v e dyadically,

Η

let

χ

be a p o i n t

let

such that

set

of

number

t

is

of

of

X ,

X.

the

Then

D Q , . . . , . . .

c a n be

to

in

Write

(i

ε I

a uniform

X - Η .

°

D = D^ be

= 0,1,...)

expanded

thus

t = ag + a ^ / 2 + . . . + a ^ / 2 where e a c h

D

X

c o n s t r u c t i o n due

be a c l o s e d

inductively,

real

that

an i n g e n i o u s

seme e n t o u r a g e

proceeding

entourages

of

Let

and

for

we s u p p o s e

numerator

a.

is

+ . ..

either

0

or

1 .

Consider

the

1

subset t

e I

I' for

of

I

consisting

which the

dyadic

associate

an e n t o u r a g e

E^.

follows;

if

a^ = 1 ,

in the expansion

i ^ , i j , . . . , ijj

of

dyadic

expansion

rationaIs,i.e.

terminates.

of

X

are

the indices

of

t

with each such

,

with

i^

i

for

< i2

those

We t

as which . ..

i^^ ,

then E. = D. t i

η

l

«D. ,

I n a d d i t i o n we s e t I assert

that

n-1

<>...»D. 1, 1

E^ = ΔΧ , if

s,t

ε I'

the

diagonal.

with

s < t

then

E^ c e ^ .

.

The uniform topology This

is

o b v i o u s when

(k+l)/2n

,

a r e of

case

now f o l l o w s

t h e form

for

We now d e f i n e = infit X ,

ς i EgCxJ remains

In f a c t ,

ζ

< s

α

is

α(ξ)

if

CK/2" ,

a few s l i p s

[10].

in

s t

The g e n e r a l

.

index

i

numerator

in a^

. Κ

by

If

ζ

ε Η

α(ς)

= l

then .

it

continuous. For l e t

(ζ,η)

e D^

η

be a

then

t =

{k+l)/2'^

c Ε.[χ] t-

,

Thus t h e

and c o m p l e t e s

the opportunity of

while

α(η) < s

interval

an i n t e r v a l

and s o t h e a s s e r t i o n

the proof

then

since

similarly.

take

α : X

and

,

» Ε^)[χ] s

continuity

I could

i = 2"t

that if

s = k/2"

,

> k

which the

continuous.

cannot contain

(k+2)/2"]

establishes Perhaps

S t

c a s e when

.

η

α(η)

,

is

α

uniformly

a(n)

S

and

for

and s o

that

I assert

ε Ε [ χ ] => η ε α(ζ)

t

Hence,

special

with

= 0 .

- α (η) I < that

,

: ζ ε E^Lx]} a(x)

is

in the

t o be t h e g r e a t e s t

and

E^^ = ΔΧ ,

implies

implies

η

k/2"

D^^ = E^ <= E^ .

·

k = 2"s

e ]R'

integer;

For o b s e r v e α (ζ)

s

and by

since

|α(ζ)

then

the function

t o be s h o w n

positive

t h e form

k/^n,

and s e t t i n g

ζ /

since

by t a k i n g

binary expansions

when

a r e of

we o b t a i n t h e a s s e r t i o n

s,t

a(S)

t

respectively,

by i n d u c t i o n ,

non-zero,

s,

of

the

between form

follows.

the proof

of

to mention

the corresponding

This (3.26).

that

result

there (11.5)

are of

th( is

Completeness

Recall

that

sequence

in a metric is

said to

be a C a u c h y s e q u e n c e , integer

k

and completion

if

space

X,

satisfy

with metric

^ ^

for

the condition

exists

an i n t e g e r

m,n > k .

being k

move s t r a i g h t Definition satisfies

such t h a t

As u s u a l ,

sequences that g i v e on

(4.1).

t h a t for

however,

'^m'^n' it

Λ filter

Thus a s e q u e n c e

F

in

D X

F

of

of

the discrete

whereas in the c a s e

filters

satisfy in

a s a Cauchy f i l t e r .

c o n d i t i o n t o be s a t i s f i e d

the

satisfies

of

if the

uniform

t h e c o n d i t i o n are the t r i v i a l

the

uniform

condition.

(4.1)

is

satisfied

Clearly

it

is

for

X

a D-small

elementary f i l t e r

In t h e c a s e

When t h e c o n d i t i o n

Proposition

and s o we

t h e Cauchy c o n d i t i o n

filters,

structure

than

X.

principal

form

there

space

contains

the o n l y f l i t e r s which s a t i s f y

F

theory,

on t h e u n i f o r m

structure

structure a l l

obvious

^^^

i s f i l t e r s rather

satisfies

the corresponding

Cauchy c o n d i t i o n .

an

The

spaces in the D

we d e s c r i b e

sufficient

for

the

b a s i c e n t o u r a g e s when t h e

uni-

i s g i v e n by a b a s e .

(4.2).

to

to

each entourage

and o n l y i f

m,n a k.

^ ^

t h e more s a t i s f a c t o r y

and

there e x i s t s

each entourage

t h e Cauchy c o n d i t i o n i f

member f o r

e

all

Cauchy c o n c e p t c a n be e x t e n d e d t o u n i f o r m way,

a

the Cauchy c o n d i t i o n ,

for each p o s i t i v e

such that

p,

Let

F

be a f i l t e r

on t h e u n i f o r m

space

Completeness and completion X.

If

F

a Cauchy

i s convergent

S

in the uniform

this,

first

observe

that

any r e f i n e m e n t

filter

is

a l s o a Cauchy f i l t e r .

If

X

X

then

of

F

is

a refinement

and s o t h e r e s u l t w i l l i s a Cauchy f i l t e r . exists

follow

F

Ε

such t h a t

Proposition

(4.3).

space

Then e v e r y a d h e r e n c e p o i n t

F

entourage

Μ

of

X Μ

such that

F

E[MJ then

and s o

X

D-small,

as

of

F

W

there

Ε ο Ε c D, and

then

required.

be a C a u c h y f i l t e r

X

Μ

e Μ

of

of

and

so

let

Since

on t h e

is

F.

Ε

F

uniform

also a

limit

be a

symmetric

i s Cauchy

there

Then χ

is

Μ c D[x].

a l i m i t p o i n t of

The Cauchy c o n d i t i o n

X

If

X M C E O E C D .

χ is

D

Ε ° Ε c D.

an E - s m a l l member C

is

of

that

F.

For g i v e n an e n t o u r a g e

exists

D

point

filter

o n c e we h a v e p r o v e d

neighbourhood Let

a Cauchy

t h e neighbourhood

the uniform

X.

E[x]

of

c o n v e r g e s t o the

But g i v e n an e n t o u r a g e

a symmetric entourage

p o i n t of

is

filter.

To s e e

of

topology then · F

F,

an a d h e r e n c e Therefore

as

point

D[x] < F

asserted.

is particularly

important

in

the

t h e o r y of f u n c t i o n - s p a c e s . To i l l u s t r a t e t h i s , c o n s i d e r y set Y of f u n c t i o n s X -·• Y, w h e r e X i s a s e t and Y

the is V

a uniform

space.

There

a n a t u r a l uniform uniform

product

power of

Y.

structure

of

i s more t h a n o n e way o f g i v i n g

structure.

structure,

One p o s s i b i l i t y X Y

regarding

In t h i s c o n t e x t

this

is

pointwise convergence,

is

too coarse

instead

to

structure

used.

known a s t h e

pointwise

of

pointwise

of uniform

This i s t h e uniform

the

as the uniform

X

th

uniform uniform

convergence. convergence

c a p t u r e many i m p o r t a n t r e l a t i o n s h i p s

the unifor m s t r u c t u r e

frequently

to use

and t h e a s s o c i a t e d

t o p o l o g y i s known a s t h e t o p o l o g y of However t h e u n i f o r m

is

Y

convergence

and is

structure generated

by

52

.Μ. James

the family

of

subsets

entourage

D

of

where

: Χ ->• Υ,

θ ,φ

uniform

uniform

if

where

is

(θχφ)ΔΧ c D.

separated

For

of p o i n t w i s e

Y^

is

with

Y^,

by

and w i t h

(θ,φ), associated

Y^

in the

(2.9),

structure

i s being used.

convergent

filter

convergent,

and s i m i l a r l y

for

either

uniform

and s o i s

s u b s e t s of

structure

separated

convergence. Y^,

and

it

is

associated

Thus a s e q u e n c e

or

Cauchy,

filter

or may be

but not uniform l y c o n v e r g e n t .

a uniformly

convergence.

in

of uniform

may be p o i n t w i s e Cauchy b u t n o t u n i f o r m l y pointwise convergent

each

of uniform

separated

t o be c l e a r which uniform

topology

for

The

then so i s

convergence,

the uniform

D,

of p a i r s of f u n c t i o n s

such that

Y

When d e a l i n g

uniform

Y^ χ Y^

i s known a s t h e t o p o l o g y

refinement

essential

of

consists

structure.

structure in i t s

Y,

topology

Note t h a t

D

Obviously

i s uniformly

Cauchy and

sequences.

In t h e

pointwise

other

d i r e c t i o n we h a v e Proposition where

X

is

uniformly

in <φ^>

Φ,

a set

Let and

φ

of

(4.5) .

Φ Y

Cauchy f i l t e r

s a n e member

Corollary

(4.4).

Φ

elementary f i l t e r

to

follows

F

by a p p l y i n g

let

D

Since

π

to

converges

to

Y,

F

be a

to

π^ (Φ)

φ.

Then

the p r o p o s i t i o n

Ε

of

Y

Let in

θ e Μ Y,

and l e t

the

To p r o v e

of

such

to

Y. that

i s u n i f o r m l y Cauchy t h e r e e x i s t s F.

to

sequence

φ.

be any e n t o u r a g e

Ε ο Ε c D.

of

Cauchy

with the sequence.

a symmetric entourage

Μ

X

φ.

There e x i s t s

F

Let

converges uniformly

associated

itself,

E - s m a l l member

space.

which c o n v e r g e s p o i n t w i s e

F

functions

be a u n i f o r m l y

converges uniformly The c o r o l l a r y

of

which c o n v e r g e s p o i n t w i s e

Then

Let

as above,

the proposition

i s a uniform

on

Φ.

be a f a m i l y

χ £ X.

and s o t h e u n i f o r m

an Then

neighbour-

Completeness and completion hood

of

ξ £ M.

Now

ττ^ίφ)

E-anall,

meets

π^^(Μ)

in

ττ^ ( ξ ) ,

e Ε,

since

θ,ξ

e Μ

(π ( θ ) , π ^ ( ς ) ) Λ

while

5 say,

and

where

Μ

is

Χ

{π^{ξ),7Γ Λ

(φ))

e Ε

by c h o i c e of

ξ.

Therefore

X

(π^ (θ) , π ^ (φ) ) £ Ε ο Ε C D . In o t h e r w o r d s

(Ο,φ)

for

we o b t a i n

all

θ £ Μ

Therefore

F

Proposition function,

(4.6).

Cauchy f i l t e r

^E

is a set

Then

on φ*6

is Ε

φ^F

of

which i s

spaces.

Y,

the

X.

is

true

ε F.

asserted. continuous

If

a Cauchy f i l t e r

of

F

as

0[φ]

be a u n i f o r m l y

are uniform φ

Μ

φ,

this

F

on

is

a

Y.

preimage

Since

F

is

Cauchy

which i s D-small. E-small.

The

Therefore

image φ

asserted.

Y

Y

Y

an e n t o u r a g e o f

(4.7) •

and

Y

then

a member

as

Proposition

filter

X

a member o f

i s Cauchy,

and

Since

and h e n c e

to

φ : X

an e n t o u r a g e is

there e x i s t s is

0 e 0[φ].

Μ c ϋ[φ]

Let

X

on

For g i v e n

φΜ

i.e.

converges uniformly

where

D = (φχφ)

£ D,

Let

φ : X ->• Y

i s a uniform

such t h a t

satisfies

φ*6

be a f u n c t i o n ,

space.

Let

i s defined

G

where

be a Cauchy

as a f i l t e r

t h e Cauchy c o n d i t i o n

X

in the

on

X.

induced

uniform

structure. For subset

let

Ε

of

Y

D = (φχφ)

^E.

be any e n t o u r a g e

then

φ ^N

is

of

a D-small

Since the preimages

through the entourages s t r u c t u r e we o b t a i n

of

Y,

Y.

If

subset

(φχφ)

generate

Ν

is

of

^E,

an E - s m a l l

X, as

the induced

where Ε

uniform

(4.7).

The Cauchy c o n d i t i o n c a n be u s e d t o c h a r a c t e r i z e bounded u n i f o r m Proposition i f and o n l v i f

spaces,

(4.8).

runs

as

follows.

The u n i f o r m

every f i l t e r

totally

on

X

space

X

is

totally

admits aCauchy

bounded

refinement.

ι.m. James For s u p p o s e t h a t F

on

X

let

existence totally

F'

of

is

totally

b e an u l t r a f i l t e r

F'

follows

frctn

bounded t h e r e e x i s t s ,

finite

covering

of

ultrafilter

one of

and s o

is

F'

Then t h e r e a proper

X

subset

for

X

for

generates

G.

X - OLSj

meets

X - D[S]

s u c h an Μ C DLX]

S

F

and s o

and h e n c e

proof

of

Corollary and o n l y

to

if

χ

Τ

a metric

Euclidean metric

(this

of

B.

subset

by

is

on

is

of

X.

As

family

of

I

assert

subset

exists

and s o

at

is

Choose Then

where

totally

X

is

said is

sequentially

is

not.

and

the

t o be

sequentiallv

convergent. ccmplete

We e x t e n d spaces,

if

filter.

in

number s y s t e m ) ,

t o uniform

to

bounded

a t h e o r e m o r an a x i o m

the real

X,

once

space

X

of

of

X - D[T] b e l o n g s

X

in

Μ

S

G.

are d i s j o i n t ,

a Cauchy

t h e term s e q u e n t i a l l v c o m p l e t e

is

the i n t e r s e c t i o n .

is

rationals

,

D[S]

that there

X

to define

an

(Λ.5)

of

X.

and h e n c e of

space

either

a

bounded.

S

on

is

X,

F'

such t h a t

F

We d e d u c e

e a c h Cauchy s e q u e n c e

S

X

each f i n i t e

F

finite

each u l t r a f i l t e r

on t h e m e t h o d u s e d

of

the corresponding

X - DLT]

The u n i f o r m

line

subspace

is

is

C h o o s e a D - s m a l l member for

i s complete.

that

F',

X

Thus we o b t a i n o u r c o n t r a d i c t i o n

example the r e a l

the

F.

(the

refinement.

be a p o i n t

and

G.

(4.9) . if

of

a member o f

Μ

But

(4.8)

Recall cOTplete

is

and l e t

Τ = S υ {χ}.

Since

to obtain a c o n t r a d i c t i o n , G

D

not t o t a l l y

a filter

a Cauchy r e f i n e m e n t

F

Since

to

filter

Cauchy.

of

subsets

d o e s n o t admit a Cauchy

since

is

D

that

Μ

is

X

X - D[S]

Then

lemma).

subsets.

F

subsets

For s u p p o s e ,

of

every finite

runs through the f i n i t e

F

refinement

belongs

Hence

an e n t o u r a g e

of

Given a

each entourage

these subsets

suppose t h a t

exists

bounded.

Zorn's

by D - s m a l l

Cauchy.

Conversely,

S

X

For the depending whereas

the u s e in the

of obvious

Completeness and completion w a y , and t h e n Definitio n

Intrcxluce

(4.10).

Cauchy f i l t e r

on

The u n i f o n n X

is

uniform

t r i v i a l uniform

filter

with a g i v e n

the converse holds, Proposition

(4.11).

only

is

it

as

sequentially that

each

complete,

rather d i f f e r e n t

sequentially

to the

sequence.

reasons.

elementary

For m e t r i c

spaces

space

that

whereas

complete

the property

X

is canplete

the r e a l

line

IR

t h e open i n t e r v a l

and s o n o t c c m p l e t e .

of

if

(and

ccmplete.

exeimple,

in the Euclidean metric,

fore,

for

are always

s p a c e s are a l s o

The m e t r i c

for

if

in

sequentially

T h i s shows,

not

i s complete

s i n c e we c a n a p p l y t h e c o n d i t i o n

associated

if)

X

spaces

spaces,

Obviously c o n p l e t e uniform canplete,

space

convergent.

For e x a m p l e d i s c r e t e and s o a r e

5i

canpleteness

is

complete,

(0,1)

We s e e ,

i s not a

is there-

topological

invariant. To p r o v e metric on

space,

X.

with metric

e-small,

f. Μ.

So f o r

in the

M^ ry

^ ...

of m a n b e r s o f F

P,

F,

e

that

^ Mj^ ^

be a Cauchy

ρ(ζ,η)

a member < e

η

filter Μ

of

F

whenever

we may s e t

e = 1/n

w i t h M^^

... l/^^-small.

^

ζ

S i n c e e a c h member

a point

ζ^^

of

M^^

M^

and

w h i c h s a t i s f i e s t h e Cauchy

t o seme p o i n t then

ζ

of

^

C h o o s e an i n t e g e r

m > 2n

such t h a t

ρ(ζ,ξ_)

Now f o r

any p o i n t

.

complete

sequence

and s o c o n v e r g e s

< l/2n

F

integer

i s n o n - e m p t y we may c h o o s e

a neighbourhood of

is a sequentially

there e x i s t s

sense

o b t a i n a sequence dition

X

and l e t

each p o s i t i v e

and o b t a i n a n e s t e d

of

suppose t h a t

For e a c h p o s i t i v e

which i s ζ,η

(4.11)

X.

If

U

conis

seme i n t e g e r e

χ e M„

then we h a v e

n.

I.M. James ρ(ζ^^χ)

< 1/m

ρ(ς,χ) χ e U

so

is

i s conpact

M^ c υ .

(4.12). if

Also

compactness

in

G

point.

if

X

i s c c m p l e t e and t o t a l l y

that

X

X

X

is

.

totally

X

F

that

X

is

F

Consequently

convergent.

Proposition

and s o

F

on

each

X

is

from t h e d e f i n i t i o n

If

of

F

a Cauchy r e f i n e m e n t X

F

is

This c o m p l e t e s

D[x 1

adherence

the

space.

i s complete

G

complete.

a d m i t s an

be a u n i f o r m

such t h a t

F .

DCx]

Then

since

D[x]

Therefore

(4.14) · where

X

Let and

on

on

F

D[x]

i s complete, X

X ,

Μ c D[x]

i s a member o f

i s a Cauchy f i l t e r

convergent,

anbeddina,

X

be a C a u c h y f i l t e r

D - s r a a l l member o f

D[X]

filter

t o t a l l y bounded.

compact.

Let

X

bounded.

i s Cauchy t h e n

bounded,

X

proof.

Let for

D

all

be

χ c X

ccmplete.

For l e t

on

and

coverings.

is

(4.13).

Then

and s o e v e r y

a d m i t s a l i m i t p o i n t and h e n c e

is

space.

Further suppose t h a t

Therefore

X

If

then there e x i s t s

(4.8)

an e n t o u r a g e o f Then

i s compact,

point.

terms of

on

Proposition

F

required.

be a u n i f o r m

Conversely suppose

Then

i s a member o f

i s a l i m i t p o i n t by ( 4 . 3 ) ,

complete.

by

as

U

X

and o n l y

adherence p o i n t

a filter

.

Thus

F

so

Let

a d m i t s an a d h e r e n c e

F ,

< 1/n

and

t o t h e g e n e r a l c a s e we p r o v e

For s u p p o s e

of

l/m-small,

a l i m i t p o i n t of

Proposition

is

is

+ (1/m)

and s o

Returning

X

M^

< {l/2n)

Hence ς

since

for

Y

Y

each

Μ χ

be a

c Μ .

and t h e t r a c e o f .

The t r a c e

and s o

i s complete,

ψ : X

and l e t

as

F

F

itself

is

is

asserted.

be a c l o s e d

a r e uniform

of

F

spaces.

uniform If

Y

is

Completeness and completion complete

then so i s

For l e t

F

Cauchy f i l t e r If

y

be a C a u c h y f i l t e r

on

Y

and s o i s

i s a l i m i t p o i n t of

then is

X.

y £ φΧ

since

an a d h e r e n c e p o i n t o f In p a r t i c u l a r

space

Proposition

Let

Then e a c h c c m p l e t e

subspace

such that

in

X

i s a Cauchy f i l t e r

Nj^

on

X

is

χ /

to

of

l^t

X

i s ccmplete

converse,

let

members of

F

F

on

χ

A.

that

G

A

on

A

A

i s a non-

W^

a = χ

,

is

by

is

always

a

(Λ. 1 1 ) ,

of the

extension

on

X.

trace

However,

the

by s u b s e t s o f

of

consider

t h e form

an e n t o u r a g e o f

X

and

Since

F

is

a refinement

G

i t w i l l be s u f f i c i e n t

converges.

D[M],

i s a member of

Now e a c h s e t

A

T h e r e may be

and s o t h e

Μ

on

To p r o v e

is

of

since

the uniform

D

G

would

convergent.

and s o c o n v e r g e n t .

generated

of

contradiction.

and o n l y i f

A,

χ

ccmplete

where

c a n show t h a t

of

t h e e x t e n s i o n o f a Cauchy f i l t e r X

point

the trace

But t h e n

if

be a Cauchy f i l t e r

X

A

be a d e n s e s u b s e t

which do n o t meet

on

uniform

space.

filter

Since

may n o t be d e f i n e d , a s a f i l t e r .

the f i l t e r

(4.3).

b e an a d h e r e n c e

a ε A.

us our

i s ccmplete

a Cauchy f i l t e r

a Cauchy f i l t e r

^(y)

closed.

Hence by ( 4 . 7 ) ,

and s o

This g i v e s

Then

X

by

a canplete

The n e i g h b o u r h o o d

p o i n t of

X,

A

A.

converges t o sane point

space

F on

is

Let

on

(4.16).

is

X.

a Cauchy f i l t e r

Proposition

If

X

X.

Hausdorff.

X

χ=φ

be a s e p a r a t e d u n i f o r m

on

b e an a d h e r e n c e

canplete.

point,

point,

s u b s p a c e of

is a

is

and h e n c e a l i m i t

a contradiction

of

A

the trace

Y

Thus

s u b s p a c e of

of

is

since

is closed.

X

For s u p p o s e t o o b t a i n

Λ

φ^Ρ

I n t h e o t h e r d i r e c t i o n we h a v e

(4.15).

closed ccmplete

φ

each c l o s e d

is canplete.

Then

h e n c e an a d h e r e n c e

and F

X.

convergent,

Φ^,F,

X ε F

on

D[M]

is

F. if

we

a neigh-

. Ό

I.IVI.JURNA

b o u r h o o d of

Μ

as a f i l t e r

on

and s o m e e t s A,

where

A.

Therefore

a : A c X.

o*G

Since

is

G

is

defined a Cauchy

*

filter

so i s

a*G,

by

(4.7),

and h e n c e

so i s

σ^,σ G,

by

*

(4.6).

Therefore

σ*σ G

i s convergent,

by t h e

condition,

*

and s o h a s an a d h e r e n c e p o i n t . of G

G,

which t h e r e f o r e

But

uniform

spaces.

For Then

(4.17).

let

π .J * F

is

so converges Therefore

a Cauchy f i l t e r

as

(4.l6).

let

F

be a f a m i l y

to

on

But

of

ccmplete

HX^

is

on t h e u n i f o r m X.J

Xj

of

x,

where

for

Xj,

also

product.

each index

since π j (x)

Xj

= χj ,

j,

is

structure

X

Let

Y

the

set

ccmplete

be a c o m p l e t e u n i f o r m y Y

of

functions

we m e a n ,

of c o u r s e ,

of u n i f o r m c o n v e r g e n c e .

be a u n i f o r m l y

Cauchy i n

to

Y

for

φ (x) ,

Cauchy f i l t e r

each p o i n t

say.

the function

and s o

subset

froTi Φ

X

IlXj

X

space. Y

is

complete

To p r o v e γ Y .

on

to

Φ

φ e

Y,

i s uniformly

of

χ

of

X

In o t h e r words, thus defined

converges uniformly to φ. Now s u p p o s e t h a t X is Y the

and

complete.

Then

Y

π

w i t h t h e uniform closed,

i.e.

F

converges

and s o ,

by

topology.

closed

in

F

Y^

is to

pointwise

(4.4) , F

space.

of c o n t i n u o u s

the

*

and s o c o n v e r g e s

a topological

consisting

in

(4.l8),

*

a limit

complete.

complete.

By u n i f o r m l y uniform

refinement

asserted.

each s e t

uniformly

a

(4.3).

be a Cauchy f i l t e r

converges

is ccmplete,

Then f o r

(Xj)

t o seme p o i n t

F

Proposition

Let

by

Then t h e u n i f o r m p r o d u c t

F

is

h a s t h e same a d h e r e n c e p o i n t .

i s C a u c h y and s o c o n v e r g e n t ,

Proposition

σ,ο G

Consider

functions

I assert with the

that topology

of u n i f o r m c o n v e r g e n c e . For l e t φ : X Y be a f u n c t i o n w h i c h f a i l s t o b e c o n t i n u o u s a t some p o i n t χ of X. Then

Completeness and completion for

seme e n t o u r a g e

contains

of

Y

no n e i g h b o u r h o o d

entourage of ψ € ΕI φ ]

D

Y

φ ^(0[φ(χ)])

x.

If

is

r; c ε

» Ε

of

such that

we h a v e

the s u b s e t Ε D

a

"of

X

symmetric

t h e n for

each function

that

ψ"^ (EL Ψ(χ} ] c φ"^ (ο[φ (χ) J) and s o c o n t a i n s n o n e i g h b o u r h o o d o f neighbourhood

of

are d i s c o n t i n u o u s

φ

in

at

asserted.

We d e d u c e

Proposition

(4.19).

Then f o r

X -»· Y

is

Then

consisting

x.

Therefore

Let

Y

each t o p o l o g i c a l

functions

x. of

Φ

Είφ]

is a

functions

which y

is closed

in

be a c o m p l e t e u n i f o n i i

space

X

uniformly

the

set

Φ

of

Y ,

as

space. continuous

ccmplete.

For is

unifonnly

complete,

by

(4.18),

and s o

each

γ

uniformly c l o s e d

subspace

of

Y

i s uniformly

complete,

by

<4.14). Q u o t i e n t s of in general. example,

(4. 20)•

Let

on t h e u n i f o r m classes

i s ccmplete

then so i s

To s e e

this

be an u l t r a f i l t e r

condition

For l e t

there e x i s t

R

are not

canplete,

i s required

let

space

X.

as,

for

F

is

H*F,

I assert

°R.

bounded.

that

Now

F

E, F

space on

where G

be any e n t o u r a g e of

symmetric entourages

F " R ^ R ' S ' ^ D

totally

be a Cauchy f i l t e r of

equivalence

S u p p o s e t h a t e a c h of

the q u o t i e n t uniform

refinement

D

be a c o m p a t i b l e

RLXJ(X€X)

the natural projection. X.

spaces

in

equivalence

on

uniform

Some a d d i t i o n a l

Proposition relation

canplete

If

X

X/R.

X/R

and l e t

π : X -»• X/R is

X. of

the

a Cauchy By

X

G

is

filter

conpatibility

such

that

i s Cauchy and s o c o n t a i n s a

bU

/.Μ. James

("Χπ) F - s m a l l member Μ and

say.

If

χ

{ (πχπ)Γ) [π(χ) ] c π(Ε[χ])

« ττ ^M

then

,

hence c

(RoE)[xJ

Since

RCxJ

finite

subset

(D°DKS] is

Μ,

is

totally S

E G

c DCRLX]] .

of

bounded,

X

and s o

by h y p o t h e s i s ,

such that

RixJ

(D-D) [ Ζ ] E G,

an u l t r a f i l t e r .

Therefore

G

= DCSJ.

for is

there e x i s t s

some

We h a v e

Ξ E X,

a

that

since

a Cauchy f i l t e r ,

G

as

asserted. Now s u p p o s e

X

on

1T*F.

By w h a t we h a v e p r o v e d ,

π(χ) F,

is

X

and l e t

i s complete.

filter

complete

X/R

that

G

point

G

of

is

is

F,

refinement

x,

say.

s u b s e t of

with the

same

(4.21).

space

X.

Let

where

Y

exists

a uniformly

ψI A = φ ,

since

F

on

X

then

πΛ

is

and

Let

A

of

Cauchy.

if

A

is

s u b s e t of

Y

be a d e n s e s u b s e t o f

be a u n i f o r m l y

ψ ψ

continuous function is

a

X/R,

by

the uniform

continuous

function,

space.

Then

there

ψ : X ->• Y

such

that

unique.

at a given point

The n e i g h b o u r h o o d f i l t e r

A,

Therefore

that

i s a ccmplete

a separated complete uniform

Cauchy c o n d i t i o n filter

the

ccmplete.

φ : A

To d e f i n e follows.

on

hypothesis.

proposition

is

of

an a d h e r e n c e p o i n t

The same a r g u m e n t s h o w s , more g e n e r a l l y , ccmplete

be a Cauchy

a Cauchy f i l t e r

hence

and h e n c e a l i m i t p o i n t o f X/R

F

be an u l t r a f i l t e r

and s o a d m i t s a l i m i t p o i n t

a limit

Therefore

Let

and s o t h e t r a c e (4.7).

Since

χ N^

of

X

of

we p r o c e e d

χ

satisfies

of

on

A

φ

i s uniformly

is

as the

a Cauchy

continuous.

Completeness and completion t h e image o f

the trace

Y

is canplete

y

i s unique

is

Y

is

N e x t we show t h a t on

A.

since

φ

is

neighbourhood V e G^, ψ (x)

of

U

since

of

of

Y

x'

D

η Ν'

is

N'

of

Φ V

is

a n e i g h b o u r h o o d of

χ

in

for

φ (UnA) c V

G^

converges

sane

and so to

Ν

contained

into

ψ

on

A

y

and

Ε .

X

F.

F,

be a n y that

X that

This w i l l

establish

to

(x ,x ')

x, x ' ,

Thus f o r

any

But s i n c e

M' £ F' Φ*F

φΝ ^ Γ ί ψ ( χ ) ]

φΝ' c Γ [ ψ ( χ ' ) ] points

£ D.

respect-

Μ T F,

we h a v e t h a t

Similarly

so

of

I assert

φΜ x φΜ' c f .

the

D

such that

converging

and s o

F'.

Ε

for

ξ £ Μ η Ν

pairs

(φ(ζ·),ψ(χ·))

that

£ F ο F " F c Ε ,

required. Finally,

so

X.

(φ(ζ),ψ(χ)), in

F.

open t h e r e e x i s t members

of

of

let

an o p e n e n t o u r a g e

F •> F ° F

on

we h a v e t h a t

( ψ ( χ ) , ψ ( χ ' )) as

= y .

be a s y m m e t r i c e n t o u r a g e s u c h

by d e f i n i t i o n

(φ(ς),Φ(ς')), are a l l

χ e Α.

continuous,

be p o i n t s o f

ψ (x)

s a n e member

£ Μ'

F

into

Μ X M' ^ D

some member ς'

D

be f i l t e r s

converges to

and

If

Then

D η (AxA)

F,

such that

for

maps

X,

Since

Ψίχ)

φ ^V = U η A

Thus

There e x i s t s

So l e t

ively.

X.

i s uniformly

continuity

F'

in

and l e t

maps

the uniform

Let

ψ

φ χ φ

ψ χ ψ

is

G^

.

F ο F ο F c E. such t h a t

χ

of

'Since

c o i n c i d e s with

where

and s o

φ (UnA) c G^.

To show t h a t entourage

φ '''V

Y.

We d e f i n e

y = φ(χ),

then

on

y e Y

thus d e f i n e d ,

continuous,

= y = φ (X)

then

ψ,

y

G^

a limit point

separated.

For s u p p o s e t h a t

a neighbourhood A,

a Cauchy f i l t e r

there e x i s t s

since

61

we h a v e t o e s t a b l i s h u n i q u e n e s s .

In f a c t

if

and

I.M. James ψ'

is

a continuous extension

continuous, since

Y

then

is

Corollary

( 4 . 2 2) .

ively.

If

uniformly

Aj^

is

extension

of

respect-

equivalent

and s o

c a n be c o n s t r u c t e d

P,.^

X^, then

X^

f

: X^^

X2,

extends

by

gf

gf

= id

is

,

g : X j "*"

: X^^ -»• Xj^ id^^

is

on

·

an

X^

is

also

Similarly

equivalence,

the c o l l e c t i o n

a

while

by u n i q u e n e s s .

a uniform

to

(4.21),

continuous function

The i d e n t i t y

f

02»

If

as

asserted.

of Cauchy f i l t e r s ,

are c a l l e d m i n i m a l Cauchy f i l t e r s . as

is

the inclusion.

then

continuous function

X,

to

denote

equivalence

t o a uniformly

space

X^ ,

•

Minimal e l e m e n t s of jniform

X.

s u b s e t s of

to

Therefore

on

A,

be d e n s e

σ^^α ^α = σ^ .

.

X2

on t h e c l o s u r e of

uniform

a uniform

s u c h an e x t e n s i o n

uniformly

be s e p a r a t e d c c m p l e t e

i s uniformly

Now t h e u n i f o r m l y

necessarily

X^ ,

continuous function

extends

not

coincide

σ^ : A^ - • X ^ ( i = l , 2 )

A2

uniformly

φ,

and s o c o i n c i d e

Λ2

equivalent

α : Aj^

ψ'

Let

A^^ ,

For l e t

Eg = i d

and

Hausdorff,

s p a c e s and l e t

Oj^a ^

ψ

of

on a

These

follows.

Proposition

(4.23).

space

Then t h e r e e x i s t s o n e and o n l y o n e minimal Cauchy f i l t e r

FQ Fq

X.

on

X

Let

such that

is given

F

f

i s a refinement

by t h e s u b s e t s

through t h e e n t o u r a g e s of of

be a C a u c h y f i l t e r

DCM]

of

and

Μ

X

of F,

on t h e u n i f o r m

FG. where

A base D

for

runs

r u n s t h r o u g h t h e members

F. As i n

FG

on

Fg

is

X.

(4.16), If

Μ

of

i s D-small

then

a Cauchy f i l t e r

the proof

suppose t h a t

G i v e n an e n t o u r a g e exists

the family

D

a D - s m a l l member

subsets

D[M] g e n e r a t e s

D(;M]

X,

G

is

a Cauchy f i l t e r

X

and a member

Ν

of

G

and

coarser

Ν

filter

( D » D » D ) - s m a l l and

on

of

clearly

is

a

Μ

than

F.

refined of

meets

by

so

To c o m p l e t e F.

F

there

M,

since

Completeness and completion Ν e F,

hence

a refinement

Ν c D[MJ of

Fg

members o f

F ,

For e x a m p l e

take

given point filter

FQ

F^

it

is

is

χ

Then

and s o we Let

N^

is

X

sufficient

to use only

i s generated

F^

'G

is

is

basic

by a g i v e n

filter

base.

generated

just the

by a

neighbourhood

obtain be a u n i f o r m

space.

a m i n i m a l Cauchy f i l t e r

for

The

neighbour-

all points

χ

X.

Corollary uniform

( 4 . 2 5) .

space

interior

of

Let

X.

If

is

Μ

be a m i n i m a l Cauchy f i l t e r is

an e n t o u r a g e

an o p e n e n t o u r a g e ELMj

F

a member o f

open s i n c e

illustrates

D

Ε c D. Ε

Our n e x t r e s u l t , one of

of

If

A;

of

any

if

F

all

i s open,

Proposition relation

X

the the

of

(3.17) ,

E[M] = DLM]

and

result. (4.2U)

above,

filters.

a uniform

space

t h e m i n i m a l Cauchy

t o obtcjin

X

and

of

G

on

information

A

not

filter

is

defined F

about

fran

tlie

trace. Let

on t h e u n i f o r m

is

the

by

w h i c h c o n t a i n s members w h i c h do

then the t r a c e

the

then

whence

A

t h e manbers of

(4.26).

lence c l a s s e s Then

X

i t may be p o s s i b l e of

there e x i s t s ,

w h i c h may b e c o m p a r e d w i t h

on

A

meet

properties

then so i s

t h e u s e s of m i n i m a l Cauchy

F

a Cauchy f i l t e r

X

Μ £ F

S u p p o s e t h a t we h a v e a s u b s p a c e

meet

F

on

M.

For g i v e n

G

Therefore

minimal.

t o be t h e p r i n c i p a l X.

(4.24) .

hood f i l t e r of

F of

of

Corollary

D[MJ e G.

when t h e f i l t e r

χ

Njj

and s o

and s o

In c o n s t r u c t i n g

63

R[x]

R

space

be a c c m p a t i b l e X,

i s ccmplete.

such t h a t

equivalence e a c h of

Suppose t h a t

the

X/R

equiva-

is

ccmplete.

ccmplete.

For l e t t o some p o i n t

F

be a Cauchy f i l t e r

τιίχΐ

4 X/R .

on

I assert

X.

Then

w^F

converges

t h a t e a c h member o f

t h e minimi

I.M.James Cauchy f i l t e r G

G

trace

of

on

argue

that the

R[x]

is

d e t e r m i n e d by R[xJ

trace

Then

a limit point

of

ς

point

is

a limit

meets

is defined. converges

complete.

F

G

ζ

of

so t h a t

Assuming t h i s

t o some p o i n t is

itself,

R[x],

an a d h e r e n c e

since

G

we g o on

F

of

to

ζ e R[x],

since

point,

hence

and

i s Cauchy.

the refinement

the

Therefore

G,

and s o

X

is

complete. To p r o v e R[x]

for

t h e a s s e r t i o n we h a v e t o

each entourage

D

By c o m p a t i b i l i t y

we h a v e

E.

converges

Since

Ti

a neighbourhood member

Ν c Μ

DCN] C D[M] completes

of

The m a i n u s e ccmpletion

the uniform If,

of

for

it

Cauch^'filters

X

Ν c

X/R

uniform

of

as f o l l o w s .

i s not

the r e s u l t

space on

pairs

such subsets on

0^(1=1,2)

entourage

for

c D~^[R[x]]

the a s s e r t i o n

is

in

is some

and

so

and

is

X X,

(F,G)

D*

of

instead, in of

but

general. the

t h e same a s i f

separated

minimal

place.

l e t us consider with

the

construction

separated,

by t h e c o n s t r u c t i o n

For e a c h symmetric

structure

F.

ττ(Ε[χ])

A similar

the s e t

the following

entourage

D

of

X

X χ X

X

I assert

forms

of

uniform let

D*

o f m i n i m a l Cauchy f i l t e r s that

a base for

on the

a

X.

T h a t t h e fam i l y f o r m s a b a s e f o r For l e t

some and

minimal Cauchy f i l t e r s

space

of

of

πΝ c TI(EL'XJ)

w h i c h h a v e D - s m a l l members i n corunon.

family

for

(RoE)[x]

were used i n the f i r s t

set

Μ

(4.26).

minimal Cauchy f i l t e r s

the

in

This proves

i s followed

Given a uniform

denote

«R

D[M] m e e t s

out u s i n g o r d i n a r y Cauchy f i l t e r s

uniform

structure.

π (x)

space which a r i s e s

however,

quotient

Then

construction,

can be c a r r i e d

to

RLxJ.

t h e proof

and member

we h a v e t h a t

F.

meets

X

R o E c D * ^

Π{χ)

of

of

show t h a t

be a s y m m e t r i c

a filter

entourage

of

i s obvious X.

The

enough. inter-

Completeness and completion section

D = D^^ η D^

D-small

s u b s e t of

D* c D* η

X

as

The f i r s t

is also is

two o f

a uniform

is

a Cauchy f i l t e r

the three

structure on

are X

then

D*

is

s y m m e t r i c by

that X

G,

X,

and l e t Let

such t h a t both

completes

Ε

F,

(F,G)

Now

G,

on

is

the proof

D

of

In f a c t

such t h a t X.

and

G.

Now

H

F

(F,G)

minimality,

X

is

function

G,

i

D*

e D*

Μ u N, Η

: X -<· X

t h a t t h e uniform

F

is

X X

functio n, of

such

and

space

X

Thus

G η H, belong

This

thus

is

H.

defined filters

entourage and

Ν c G,

by b o t h

F

since

for

a D-small

So

common

F = Η = G,

by

separated.

t h e neighbourhood f i l t e r a s we h a v e s e e n ,

by

i(x)

and w r i t e X

to

Since

Μ e F

refined

there

is

on

uniform.

where

X

is defined

structure

symmetric

as r e q u i r e d .

which i s

of

which shows t h a t

as the canonical

base

Moreover

E*.

t h e Cauchy c o n d i t i o n

D

of

be any

be m i n i m a l Cauchy

and h e n c e o f

χ

D

for e v e r y symmetric

a m i n i m a l Cauchy f i l t e r , i

e D*.

hence D-small.

F, G

satisfies

For each p o i n t of

(F,F)

since both s e t s

t h e uniform

a filter

and

the

a D - s m a l l member,

F η G

structure

let

each symmetric entourage member o f

for

Thus i f

belong t o

of

N,

the

Then t h e u n i o n s

form t h e b a s e o f

that

be m i n i m a l Cauchy f i l t e r s

M, Ν

E* » E*

that

let

(G,H)

Ε-E-small,

we h a v e

separated. X

and s o

Η

and

We g o on t o show t h a t is

so

be a symmetric e n t o u r a g e

Μ meets

Μ υ Ν

M u N e F n H

obvious.

contains

D,

E - s m a l l members

and s o

F

the third c o n d i t i o n ,

respectively.

every

construction.

Ε = Ε c D.

there e x i s t

special conditions

also fairly

each symmetric e n t o u r a g e

of

and

b o t h D^^-small and D 2 - s m a l l ,

for

entourage

a symmetric e n t o u r a g e

required.

of

To v e r i f y

65

is

~ j = i

W^

and s o a

We r e f e r x i.

I

to assert

i n d u c e d frcm t h a t o f

X

I.M. James under

i.

For l e t

t h e one hand, Μ

of

X

if

(ζ,η)

€ D.

0[ζ]

υ 0[η]

is

ζ

and

η,

be a n y s y m m e t r i c e n t o u r a g e

(ξ,η)

which i s

that

D

e

there e x i s t s

a neighbourhood of both On t h e o t h e r

hand i f

of

X.

a D-smail

ζ

(ζ,η)

On

subset

and

η,

so

« D

then

D<>D»D-small and i s a common n e i g h b o u r h o o d

and s o

(ζ,η)

e j~^(DoDoD)*.

This proves

of

the

assertion. N e x t we s h o w , and t h a t on

iX

iX of

i s dense in

t h e uniform

Cauchy f i l t e r trace

is

a t o n e and t h e same t i m e ,

the

F,

X.

neighbourhood

where

D

is

s e t of p o i n t s

χ

contained have

in

D * [ F ] r. iX = IM,

and s o over the i*(F)

on

of

X,

So l e t

where

on

X, is

of

In t h i s consisting

uniform image iX

of

F ,

where

Μ

iX,

in

are the

X

of

X

is

i s dense

i

images under

interior

i s a member o f

on

X

iX.

χ i

that

Then

Morethe

i*F

refined

X

(4.25),

filter

to the point

seen, is

space

F.

is a

by

of i*F.

so does

the

complete. space

X

X

and a

a pair

As we h a v e s e e n ,

Furthermore of

X.

i s a refinement

: X - X.

X. i

the

iX

i n d u c e d from t h a t o f in

Tims we

in

a s we h a v e j u s t

function

F.

i s dense

in

e D*,

is

iX

which e s t a b l i s h e s

continuous

χ

Μ e F, by

on

The

neighbour-

However

F = i*i*F

X,

X.

F.

converges

and s o

of

t h e u n i o n of

a s e p a r a t e d c o m p l e t e uniform

of

minimal

is

so that

i*(F)

trace

{F,i(x))

o f a D - s m a l l member o f

way we a s s o c i a t e w i t h e a c h u n i f o r m

structure iX

the

a D-small

t h e m i n i m a l Cauchy f i l t e r

converges

extension

of

such t h a t

contains

be a Cauchy f i l t e r

G

Since

uniformly

F

IM η D*[FJ

and s o F

Cauchy f i l t e r i»G,

meets

trace

D*iF]

ccmplete

the

a symmetric e n t o u r a g e

t h e D - a n a l l m e m b e r s of

D*LF]

is

or i n y e t o t h e r w o r d s s u c h t h a t

in the i n t e r i o r

p o i n t s of

of

X,

X

this consider

i(x)(xeX)

or i n o t h e r words such t h a t hood o f

To s e e

that

X, the

also

the

the

entourages

t h e e n t o u r a g e s of

X,

Completeness and completion so that alence

iX

is

a quotient

relation

uniform

(ixi)~^A(iX)

function

c o i n c i d e s with the

of

Thus we s e e

X.

equivalence, considered

the c a n o n i c a l

as a dense

X

X,

and the' e q u i v -

determined

intersection iX

2.

s p a c e of

on

R

of

by t h e the

c a n be i d e n t i f i e d ,

separated

in Section

separated

function

embeds

when X

entourages

u p t o uniform

q u o t i e n t uniform

In p a r t i c u l a r ,

canonical

space X

already

itself

uniformly

is

in

X

subspace.

We g o on t o the

that

with the

67

following

show t h a t

the completion construction

characteristic

Condition

(4.27) .

function,

where

Let X

c o m p l e t e uniform

a uniform

space.

tinuous function

property.

φ ! X ->- Y

is

be a u n i f o r m l y

s p a c e and

Y

Then t h e r e e x i s t s

ψ : X

has

Y

such t h a t

is

continuous a

separated

a uniformly

ψ1 = φ,

and

con-

ψ

is

unique. We b e g i n by s h o w i n g function

: iX

Y

that

such t h a t

c o n t i n u o u s we n e c e s s a r i l y φ(χ) for

each

have

a uniformly

ψ^Ι = φ.

In f a c t ,

continuous since

φ is

that

= limφ^M^ χ e X.

by d e f i n i t i o n , have

there e x i s t s

and s o we c a n d e f i n e

t o show t h a t

continuous.

T h i s d e p e n d s o n l y on

the

a symmetric entourage formly continuous.

t^dix))

function

Now g i v e n

Then

since

Ε

(φχφ)

^E

of

(ixi)

^D* c: d

is

of

Y,

X,

since

and

i(x!

= l i m Φ*^^^·

thus defined

an e n t o u r a g e D

i(x),

= We

unifomly

there φ

exists is

uni-

so

(ψβΧψ^) ( D * n ( i X x i X ) ) c E, as r e q u i r e d .

N e x t we e x t e n d

ψ^

frcm

iX

( 4 . 2 l ) , and o b t a i n a u n i f o r m l y

continuous

such t h a t

ώΐ = ώ .

ώ I iX = iIj _

and s o

to

X,

function

as r e a u i r e d .

as

in

ψ : X

Y

Finally

I.. James we o b s e r v e

that

ψρ

i s u n i q u e l y d e t e r m i n e d by

the unique continuous e x t e n s i o n

(a f o r t i o r i

uniformly

of

proof

of

continuous

(4.23).

if

if

and o n l y Because

its

The u n i f o r m

stated condition

of

X

and l e t X

Ε

result

is

of

X.

X,

where

so

1~^(Ε[ζ])

: X ->· X is

structure

t h i s proves

is

a finite

D

of

covering

The i m a g e s

of

by E - s m a l l {ϊ^}.

the

K^

D

is

for

often

used

To show

that

be any

entourage

Ε » Ε

subset

Ε[ςϋ

all

ζ

D. S

of

X

form a c o v e r -

form

But

bounded

compact.

Ε[ξ](ξε3)

a covering

of

i s D-small

and

e S.

t h a t of

X

X

be an u l t r a f i l t e r

let

Ε

Since

the

under

i

Since of

where

Χ

is

totally

X

by

(ixi)

K^ = i H ^ , X

i s D-small.

K^

belongs to X

the followina

F.

Let

φ : X

a

F

Y

of

there sub-

covering their

each

j,

since

is

an

Therefore

F

i s complete.

the completion construction sense.

for

X

^E-small

form

Since

i s c o m p a c t and c o m p l e t e s t h e p r o o f

In f a c t

X.

bounded

i s c o v e r e d by

i s E-E«E-sma11,

since

on

be an e n t o u r a g e o f

K^

C a u c h y and s o c o n v e r g e n t , X

is

^(Ε[ς])

s u b s e t s and s o

and s o

o n e of

F

{Η^}

{K^},

Now

i s E-small,

ultrafilter

that

totally

bounded.

i n d u c e d frcm

let

Ε ο Ε ο Ε c D .

sets.

Kj

the

sufficiency.

G i v e n an e n t o u r a g e such t h a t

i

(ixi)~^D-sraall, X

X

a finite

i s canonical.

of

is

such t h a t

neighbourhoods

To p r o v e n e c e s s i t y ,

closures

exists

Then t h e p r e i m a g e s i

X

let

be a n e n t o u r a g e

such t h a t t h e uniform

iX

is

unique

t h e term p r e c o m p a c t

sufficient,

i s compact t h e r e

ing

space

t o t h e term t o t a l l y

the

exists

ψ

This completes

separated completion

of t h i s

a s an a l t e r n a t i v e

uniform

φ^ .

the

and

(4.2 7).

Proposition

Since

extension)

φ

This

is

shows

(4.28).

is functorial

be u n i f o r m l y

in

continuous.

Completeness and completion where istic

X

and

Y

are unifonn

p r o p e r t y we s e e

unifonnly

continuous

a s shown

below.

Here

and

i

j

spaces.

that there e x i s t s function

are c a n o n i c a l .

separated canpletion

69

of

φ.

By u s i n g

the

o n e and o n l y

φ : X -<• Y

We r e f e r

such t h a t

to

φ

as

characterone jφ = φ1,

the

5.

Topological

In t h i s arise

groups

s e c t i o n we s h a l l

from s u b s e t s

g r o u p and l e t

Η

of

be c o n c e r n e d w i t h

a group.

Specifically,

be a s u b s e t o f

G.

Then

relations let

Η

G

which

be a

determines

the

relations

Lf, = t ( ζ , η )

: η ε ζΗ}

,

Rj, = { (ξ , η )

: η € Ηξ]

.

We r e f e r Rj^

to

L^^

as the r i g h t

as the

relation:

N o t e t h a t when left

and r i g h t

a l s o have

L

Η

=

Η

for

all

subsets

H, Κ

In p a r t i c u l a r

of

left

cosets,

quotient

equivalence

=

to the n e u t r a l

Η

and

they

element

to

coincide. e

relation

both Δ.

We

(RH)'^

G,

of

" and

relation,

c o s e t s , and t h e s e t

relation,

and t h e s e t

set,

that

G.

suppose that

an e q u i v a l e n c e

left

the commutative case

H^

subsets

the

d e t e r m i n e d by

reduce to the i d e n t i t y

R

«

all

called

in

reduces

relation

for

is

relation

that

H 1

L„ π

left

written

Η

is

a subgroup of

the e q u i v a l e n c e of e q u i v a l e n c e G/H.

the equivalence

of e q u i v a l e n c e

classes classes

Similarly classes

classes

is

G.

Rj^

Then are

is is

called an

are c a l l e d

right

called

right

the

Topological groups quotient induces

set,

written

an e q u i v a l e n c e

H\G. between

For p r e s e n t p u r p o s e s the notion ization

of

as

Definition

Inversion G/H

group i s

(5.1).

Λ topological

four

W£ W

then

e e W ,

(ii)

if

W£ Ν

then

w"^ £ W,

(iii)

if

W£ Ν

then

V.V c W

(iv)

if

g £ G

and

When t h e s e

t h e members of

Ν;

for

to

L^^

as a b a s e ,

this

is

G

the

c a n be

the

W runs

left

structure.

to

take

through

uniform

R^^

Υ £ N.

given

One way i s

where

called

some

structure.

as a base

instead;

Of c o u r s e i i

W r u n t h r o u g h t h e members of

structures

X JR

a

c o i n c i d e when

= (xx',xy'+y)

t o be t h e f i l t e r

U^ = { ( x , y ) e > 0 .. ζ = (£/4,0)

G

take

is

is

filter

ccnrnutati-ve.

G = JR, κ Η

w i t h t h e m u l t i p l i c a t i o n g i v e n by

( x , y ) · ( x ' , y ')

where

in

satisfying

g ^Vg c W f o r

For an e x a m p l e w h e r e t h e y a r e d i f f e r e n t

Η

G

G

W .

The u n i f o r m

Take

character-

V £ Ν .

two w a y s .

the r i g h t uniform let

on

some

t o take the right r e l a t i o n s

is called

a group

Ν

are s a t i s f i e d

in e i t h e r of

relations

The o t h e r i s

group i s

W £ W then

conditions

uniform s t r u c t u r e

the s e t

the l o c a l

introduce

conditions:

if

b a s e for

through

together with a f i l t e r

(1)

sufficient

H\G .

follows.

the f o l l o w i n g

this

and

G

t h e m o s t a p p r o p r i a t e way t o

topological

algebraic sense

the l e f t

i n the group

.

g e n e r a t e d by t h e

: ix-1I < £,

|y|

< £}

With ,

η =

(£/4,£/2)

subsets

t o be

I.. James we h a v e Π.ς"^ = ( 1 , £ / 2 )

Thus t h e

left

ε U^ ,

= (1,2)

r e l a t i o n d e t e r m i n e d by

the r i g h t r e l a t i o n determined So t h e left

r i g h t uniform

uniform

ment of

of a s u b s e t

subsets

H.W ,

where

a base for

t h e members o f

r i g h t uniform the closure

structure, Η

i n t h e c a s e of

the

the

case.

so t h a t

bourhood f i l t e r of

function are

of

for

ε > 0 .

left

any

of

the

i s not a r e f i n e -

observe G

t h a t the uniform

are generated the

Similarly

with

W.H

i n the uniform left

uniform

left

topology of

e,

topology

structure

the given while

uniform Η

or

of

H.W.

η(W.H) these

the

Thus ii(H.W) in

the

topologies

t h e same i n

filter

either

W becomes t h e

is

the

neigh-

neighbourhood

C l e a r l y both the

and t h e m u l t i p l i c a t i o n

the

i s g i v e n by and

is

gAJ = Wg

g ε G.

of

In f a c t

topology

by

i n t h e c a s e of

in p l a c e

structure.

the uniform

any e l e m e n t G •» G

H

inversion

function

G χ G -·· G

continuous. Of c o u r s e t h e p r o c e d u r e

a topological

group

group s t r u c t u r e functions e

the

case,

Ν .

r i g h t uniform

In t h i s

filter

contain

W r u n s t h r o u g h t h e members of

of

coincide,

does not

not a refinement

i n t h e c a s e of

structure,

c a s e of

is

similarly

to the general

neighbourhoods of

U^

U^ .

right.

Returning

family

structure

structure,

the

by

Uj^

/

(i)

with

in which the i n v e r s i o n

-

(iv)

topology.

We may d e f i n e

space carrying

and

a

multiplication

Then t h e n e i g h b o u r h o o d f i l t e r

so that the l e f t

can be d e f i n e d ,

w i t h the uniform

reversible.

as a t o p o l o g i c a l

are continuous.

satisfies

structure

G

is

and r i g h t

and t h e o r i g i n a l However,

it

topology

i s more

of

uniform coincides

consistent

our a e n e r a l approach t o r e g a r d t h e t o p o l o g i c a l

structure

Topological groups as d e r i v e d

from t h e u n i f o r m

structure,

r a t h e r than t h e

Other

way r o u n d . The l e f t uniform

and r i g h t u n i f o r m

structures

groups.

considered

Another s t r u c t u r e

s e c t i o n of

the l e f t

fact

in the

of

t h e o r y of

some i m p o r t a n c e

and r i g h t s t r u c t u r e s ,

two-sided structure. uniform

s t r u c t u r e s are not t h e

However,

only

topological i s the

inter-

sometimes c a l l e d

t h e p r o p e r t i e s of t h e s e

s t r u c t u r e s w i l l n o t be d i s c u s s e d

in t h i s

other

book.

In

we s h a l l g e n e r a l l y c o n c e n t r a t e on t h e r i g h t u n i f o r m B e f o r e g o i n g any f u r t h e r

few e x a m p l e s .

Of c o u r s e ,

filter

structures Again,

any g r o u p

taking

g e n e r a t e d by

G

g r o u p by t a k i n g

e;

both l e f t

consisting

and r i g h t u n i f o r m s t r u c t u r e s is

t o be

and r i g h t

sense, the

uniform

discrete.

of

G

itself;

both

left

and t h e u n i f o r m

trivial.

b" ( n = 0 , 1 , . . . )

c o n s i d e r the r e a l

with the a d d i t i v e group s t r u c t u r e ,

i s the n e u t r a l element. t h e open t - b a l l s positive reals.

The u n i f o r m

Ν

topology is

are t r i v i a l

For a c o m m u t a t i v e e x a m p l e ,

structures

in the a l g e b r a i c

We t a k e

centred at zero, In t h i s c a s e

a r e t h e same, s i n c e structure

sidered e a r l i e r ,

is

just

n-space in which

W t o be t h e f i l t e r where

the l e f t

«

the group i s

topology

the

uniform

commutative.

the Euclidean structure

and t h e u n i f o r m

g e n e r a t e d bi

runs through

and r i g h t

zero

is

j u s t the

conEuclidean

topology.

For a non-commutative example, take the general linear group

Ga(nJR)

base for arouD bv

a

can be r e g a r d e d a s a t o p o l o g i c a l g r o u p by

W t o be t h e f i l t e r

topology

G,

a r e d i s c r e t e and t h e u n i f o r m

any g r o u p

structun

w i t h t h e t h e o r y l e t us c o n s i d e r

can be r e g a r d e d a s a t o p o l o g i c a l principal

the

of automorphisms o f

By choosing a.

we c a n r e p r e s e n t t h e e l e m e n t s of (nxn)-matrices , and h e n c e t o p o l o g i z e

the it a s a s u b s e t

I.M. James of

Then

η = 1

GZ ( η , Β )

t h e g r o u p c a n be i d e n t i f i e d

group of

real

and t h e

numbers.

left

Thus c o n s i d e r where

T

t h e group

is

topology of of

a uniform

entourages Wj^ =

{0

of :

T,

(i)

consists

of

t W^,

i.e.

uniform

of

space,

with

consists

ε D

(iv)

pairs

is

again,

non-commutative

distinct. illustrations.

t h e uniform

of

multiplicative

equivalences

θ : τ

structure

topology, the

the subsets

i.e.

Τ, of

the

neighbourhood W^,

for

all

topological

of

for

group s t r u c t u r e s

R = R,,

e l e m e n t s of

(ψφ"^(t),t)

e D

just

e Τ} .

relation

(Φ,Ψ)

all

arises

self-equivalences

G for

such

that

all

t,

Thus t h e

left

rather

of

uniform

we form

are

determined

structure

the if

e D

s.

the uniform

but in g e n e r a l

The same s i t u a t i o n

topological

for

such t h a t

structure

are

a base for

Vt

The r i g h t

(ψ(s),φ(s))

convergence not.

structures

When

where

-

readily verified.

the

the group i s

G of u n i f o r m

(e(t) ,t)

The c o n d i t i o n s

,

In t h e a s s o c i a t e d

identity

D

such that

η a 2

uniform convergence,

the

with

p r o v i d e some i n t e r e s t i n g

uniform convergence.

filter

When

and r i g h t u n i f o r m

Function-spaces

Wj^

becomes a t o p o l o g i c a l group.

G

by

i.e.

right uniform structure

from t h e

than uniform

is

group

self-

equivalences . For a n o t h e r bijections Regarding uniform

Τ

: Τ

i.e.

T,

consider

where

Τ

is

as a d i s c r e t e uniform

structure

topology, for

0

illustration

of

an i n f i n i t e

t h e t o p o l o g y of p o i n t w i s e filter

of

the

subsets : θ (t)

= t V t ε S)

,

G

In t h e

convergence,

identity

of

set.

space l e t us g i v e

pointwise convergence.

the neighbourhood

Wg = {Θ

the group

G

the

associated a base

c o n s i s t s of

the

Topological groups where

S

runs through the f i n i t e

conditions

for

verified.

a topological

The l e f t

of

T.

Again

group structure

are

readily

relation

subsets

L = L

d e t e r m i n e d by

W_

WG

consists (t) all

of p a i r s = t

for

t e S.

uniform

(Φ ,ψ)

t € S,

i.e.

Thus t h e

left

uniform

The c l a s s

of

of

pointwise

invariant

neighbourhoods. group i f

neighbourhoods fact

it

is

V

since

V

if

Obviously a l l

the group

R* Κ R

groups s i n c e

for

the

left

and

interest.

they admit

a topological

small

group

G

admits a base c o n s i s t i n g

^ = V

for

all

U

of

gVg ^ c U

the conjugates

for

gVg ^

g t G . Ν

In

there

all

of

exists

g e G ,

for

all

g e G

condition.

abelian groups are

groups with d i s c r e t e

just

which the

e a c h member

such t h a t

the previous

is

= φ(t)

i s of c o n s i d e r a b l e

Ν

gVg

for

that

ψ(t)

structure

Specifically

such t h a t

t h e n t h e u n i o n of

fulfills

such t h a t

SIN

and o n l y i f

sufficient

a neighbourhood

these

such

convergence.

structures coincide

[ 2 1 ] we c a l l

SIN

G

t o p o l o g i c a l groups for

Following

i s an

S

e l e m e n t s of

all

structure

r i g h t uniform

of

the

or t r i v i a l

considered

SIN

groups,

topology.

earlier

as

are

On t h e o t h e r

i s n o t an SIN

hand

group.

We h a v e Proposition group,

(5.2).

Let

G

in the r i g h t uniform

be a t o t a l l y bounded structure.

Then

topological

G

is

an

SIN

group. For l e t V.V.V"^ c υ have

U ί .

V.S = G

By ( i v ) where

of

be g i v e n

Since for

(5.1)

g £ V.g^

is

and l e t

V c Ν

t o t a l l y bounded,

seme f i n i t e

we h a v e

j = 1 , . . . ,n

we h a v e

G

,

subset

gj.W^.g^"^

c ν

for

Then i f

W = W^^ η . . .

for

some

and

gW.g"^ c V . g j . W . g " ^ c

such

so

V.V.V"^ c U ,

that

on t h e r i g h t ,

S = {g^^ , . . .

.

j

he

seme η W^

of

we G .

W^ e W , and

g £ G

76

LM. James

as r e q u i r e d .

It follows,

bounded on t h e r i g h t and v i c e v e r s a ; Returning

it

then is

left

v i o u s l y a uniform

is

right p^

e

g ^Vg c W.

say that let

Thus

V

likewise

ρ

a uniform

equivalence since

be a

right

to the

left

The i n v e r s i o n

with

it

by

W

of

is

e

any

such

and s o

ρ

is

(p ) ^ = Ρ

.

equivalences and c l e a r l y

to the r i g h t uniform

between

G

the

left

uniform

structure.

with

the

same

structure.

on t h e o t h e r h a n d ,

with

both

g'

structure,

function,

Therefore

is

a uniform

structure

Some of

the

and

uniformly

in the e a r l i e r

are two-sided,

in the sense

that

has the p r o p e r t y

respect

left

structure

to the

uniform

if

G

then

G

t o the

r i g h t uniform s t r u c t u r e ,

versa.

if

is

For e x a m p l e ,

G

or t o t a l l y bounded w i t h r e s p e c t

with respect So f a r

t o the r i g h t uniform

being used.

to specify

The same i s

uniform

connected

structure,

are concerned,

which of

uniformly

to the l e f t

a l s o separated, uniformly

as these properties

unnecessary

separated,

sections

a l s o has

property with respect

is

is

continuous,

continuous,

uniform

t h e r i g h t uniform

G

ob-

the

i n v a r i a n t p r o p e r t i e s we h a v e c o n s i d e r e d

then

is

are

translation where

t r a n s l a t i o n s a r e uniform

is true with respect

G

topological

^

and r i g h t

equivalence

bounded.

a l s o a uniform

a neighbourhood

^ respect

totally

translation

g

left

is

left,

Although

is

i s uniformly

i s uniformly

totally

the entourages

(p^xp^) is

p^

G

translation.

denotes

and

is

G

Left

translation

we h a v e L·^ c

n e i g h b o u r h o o d of

to

since

to l e f t

G

t o t a l l y bounded on t h e

theory,

equivalence,

For i f

g e G

that if

uniform s t r u c t u r e .

so obvious,

equivalence.

that

G

sufficient

invariant with respect

element

course,

to the general

group with t h e

not quite

of

with the

and

vice

connected structure,

or t o t a l l y

and v i c e

therefore,

bounded

versa. it

is

t h e two u n i f o r m s t r u c t u r e s

t r u e of

any s e l f - i n v e r s e

subspace

is of

Topological groups a topological

group.

For what i s properties

of uniform

topological that

groups.

that

for

G

Ce ).

that F

for

t o be t o t a l l y a finite

The c o n d i t i o n

for

each

F

W of

Proposition of

of

group

G.

since

Η

for

F

as

F

left

respect

is

a filter

Cauchy i f

t o the l e f t

course,

H.K

F

subset

There

the group

H.K

of

corresponding

is

totally

e

in

G.

S.K

of

is

Then

totally

using

subset

a finite

bounded.

H.K,

subsets

bounded.

S unior

Therefore

and h e n c e

bounded,

group

G

as

asserted.

we d e s c r i b e

t h e Cauchy c o n d i t i o n

with

and we u s e t h e term when

G

are both l e f t

however,

as the

bounded

is

totally

structure,

filters

G

Μ

some f i n i t e

i s no d i f f e r e n c e

In g e n e r a l ,

is

g e n e r a t e d by t h e

Now

Τ

satisfies

uniform

distinguished

Consider

and s o i s

for

on t h e t o p o l o g i c a l

and c o n v e r g e n t

i n any c a s e . carefully

Then

bounded.

totally

Thus

Ν

W.Ξ = G.

a member

be t o t a l l y

is

some f i n i t e

Cauchy s i m i l a r l y . of

Κ

Η <= w . S

H.K C W.S.Κ c W.W.T. If

and

structure

t o t a l l y bounded s u b s e t s

W of

F.

W be any n e i g h b o u r h o o d o f

uniform

condition

e a c h member

the f i l t e r

one-

connected

t o b e r i g h t Cauchy

t h r o u g h t h e members o f

the topological

S.K c W.T

G

to the

The

such that

is

W r u n s t h r o u g h t h e members of

Η

H,

G

N.

there e x i s t s

W.F,

Let

the r i g h t of

W

(5.3).

For l e t

on

t o be separated

In t h a t c a s e t h e

where

runs

W of

of

S

for

t o be u n i f o r m l y

subset

by t h e p r o d u c t s Μ

G

general

form

Μ reduces

that for

is

and

G

bounded i s

m i n i m a l Cauchy f i l t e r

U

for

member

M.M ^ = W .

W.M

for

t h e members o f

a filter

f o r e a c h member

such that

of

these

i n t o the appropriate

The c o n d i t i o n

uw" = ϋ

there e x i s t s

spaces

The c o n d i t i o n

the i n t e r s e c t i o n

point set is

t o f o l l o w we n e e d t o t r a n s l a t e

is

right

commutative

and r i g h t

Cauchy

t h e two c o n d i t i o n s must b e

following

example

o f homeomorphismG

shows.

φ : I ->- I ,

where

.Μ. James I = [0,1] uniform G

is

denotes

structure

the c l o s e d obtained

a topological

w i t h the uniform

from

group,

uniform c o n v e r g e n c e ,

unit

structure

of

~

for

and

(^,
structure.

η > k;

_ ^-1-f Μ Φη [ ^ J

-

we s e e

the

left

uniform

structure,

where

G

topology

is of

is

pointwise

both t h e s e examples Returning

F.G

on

and i

.

^ f

^ ^ · uniform

' i s n o t a Cauchy s e q u e n c e

in

does not converge

in

Another example

G

discrete

of

F

Let

group

G.

bijections

s p a c e and

convergence.

G

with

φ : Τ

is given

I t should be n o t e d

the Hausdorff

t o the general

the t o p o l o g i c a l

Here

^

complete.

satisfy

(5.4).

Cauchy f i l t e r

.

in fact

the group

an i n f i n i t e

Proposition

n

that

i s not

properties Τ

0 < t s

=

I. - ^

^

η > 2,

Therefore

by

since

all

similar

then for

• ί Τ Τ ^ ^

for

G.

Consider

η = 2,3,...,

a Cauchy s e q u e n c e w i t h t h e r i g h t

However,

^-J-i 1 '''2η[2ΐί]

coincides

we h a v e

- Φη'^Η

is

of

.

-

I5 < t s 1

Thus

seen,

{0
k > 4/e

ΐΦη.ρί^'

topology

structure

which i s given for

«-have

while

relative

uniform c o n v e r g e n c e .

-1 + I + t ( 2 - | )

Ρ =

the

As we h a v e a l r e a d y

with respect to the

G

e > O,

]R.

and t h e r i g h t u n i f o r m

the sequence

Let

interval with

Τ, the that

condition.

c a s e we p r o v e and

Then

G F.G

b e r i g h t Cauchy f i l t e r s is

t h e base of

a

right

G. denotes

the

filter

g e n e r a t e d by t h e

products

on

Topological groups M.N

of

G,

where

Μ

r u n s t h r o u g h t h e members o f

r u n s t h r o u g h t h e members o f e

in

G

we can c h o o s e

Cauchy.

Then c h o o s e

for

g e M.

some

Μ Ν

F

M.N

is

R^,-small.

(5.5).

complete

s u b s e t of

For l e t meets

F

Η ^.M

Κ η Η ^.M F'

ε Μ

Η

Μ e F

there e x i s t s ς ε

for

all

be a c o m p l e t e

Με

F,

a filter

F.G ^

a point

on

Η

trace converges Therefore Cauchy.

Μ.Ν~^

eW.W.W,

note that

G.

let

Κ

Then

to a limit

is

as

asserted.

if

be a

H.K

is a

F'.

H.K.

Then

intersections Κ

is

compact

therefore

k

is

a

Then

converges

G ^

ξ.η

Η .

ε Μ

also

since

h,

converges to

form a c a t e g o r y

isomorphisms.

on Η

trace H.

is

The

complete.

Η

and s o h.k.

in which the

The e q u i v a l e n c e s

So

η ε Ν

the

since

h.k

to

and h e n c e

Therefore

h e H,

limit

Η on G.

For any

a r i g h t Cauchy f i l t e r point

Κ

Since

such t h a t

a r e c o n t i n u o u s homomorphisms. topological

K.

converges to

W.G

Topological groups

on

converges to

Η.0

on

Ν c Η ^.M η K.

meeti'

generates

of

are c a l l e d

we h a v e

tc· a r i g h t Cauchy f i l t e r

ξ ε Η

a refinement complete,

group

k ε K, of

where

Itself Hence

e Ν

and

and s o t h e

F'

G

extends

Ν e G,

whence

right

V = g~^Wg

(5.4);

be a r i g h t Cauchy f i l t e r

F.G ^

and l e t

is

G.

generate

and s o

F

where

η,η'

W of

(g.C'"^)

topological

of some r e f i n e m e n t

k ^

and

This proves

a d m i t s an a d h e r e n c e p o i n t

point

of

the

R^-small,

since

Ν

F.G.

Let

compact s u b s e t of

R^-small,

(C.g"^).

i s minimal then so i s

Proposition

t o be

ζ,ξ'

'and

Given a n e i g h b o u r h o o d

t o be

When

(ξ.η) . ( ζ ' . η · = and s o

G.

F

of

is F,

being

Thus

H.K

morphisms the

category

lei

I.M. James Proposition where

G

(5.6).

and

continuous,

Let

G'

are t o p o l o g i c a l

D

is

is

φ ^W o f

Of c o u r s e topological

(5.6)

groups

r i g h t uniform

shows t h a t

is

Direct products

the d i r e c t product

and

φ

left

the

G,

G'

a subspace

topological

IIGj

to

isomorphisms in either

then

uniform

on

structures

w i t h the r i g h t uniform

between left

or

with

φ

con-

t o e i t h e r uniform

groups are defined {G^}

the

G,

of

sense

is

IIG^

is

in

the groups

equipped

the

with

group.

uniform

on t h e f a c t o r s

structure.

structure.

sense.

topological

and t h e r e b y b e c o m e s a t o p o l o g i c a l structure

then

i n t h e uniform

in the a l g e b r a i c

uniform

left

G,

i s a s u b g r o u p of

G

structure.

G'.

the i n c l u s i o n

of

uniformly

uniform

corresponding

equivalences,

is

is

corresponding

element of

Η

homomorphism,

φ

e l e m e n t of

topological

When

of

the product topology

natural

of

Thus g i v e n a f a m i l y

Moreover t h e

Then

or r i g h t

embedding w i t n r e s p e c t

Κ

o b v i o u s way.

left

the neutral

topology,

In o t h e r w o r d s

similarly

groups.

the neutral

form u n i f o r m

a uniform

p r o d u c t of

of

structure.

the r e l a t i v e stitutes

W

the b a s i c entourage

neighbourhood

be a c o n t i n u o u s

any b a s i c e n t o u r a g e of

t o some n e i g h b o u r h o o d ^D

G

with respect to either

For i f

(φχφ)

φ : G'

G^,

and

Of c o u r s e ,

the

product has

the

projections

:

HG .

are continuous following

.

GJ

homomorphisms,

characteristic

Proposition groups.

(5.7).

Let

topological

group.

and o n l y

each of

if

continuous

property.

Let

φ : Η

IIGj Then

and t h e d i r e c t

{G^}

be a family

be a f u n c t i o n , φ

is

the functions

homomorphism.

of

topological

where

Η

is a

a c o n t i n u o u s homomorphism φ^ = π^φ : Η -> G^

is

a

if

Topological groups When

G

is

conunutative both the i n v e r s i o n

and t h e m u l t i p l i c a t i o n morphisms,

therefore

i s not the

case.

Let

Η

with is

structure.

left

cosets

the uniform

uniformly

G

Hence t h e

of

that

W t W reduces connected i s

the

to

that

that

f o r e a c h member

follows

i s uniformly (2.16)

that

open, (4.26)

proposition

(5.8).

Κ c Η c G

structure. t h e same

G/H

G

compatible

π : G + G/H uniform

by

elements

G/H W.H,

G/H

be

for

G/H

for

appropriate

subsets to

w".H i s

there e x i s t s

to

be

for

all

uniformly

the

.to be t o t a l l y

projection

from

G

full

s e t G fo]

bounded

a finite

to

the n a t u r a l p r o j e c t i o n

we

i n t o the

of

is

subset

S

Κ

is

G/H

from

any s u b g r o u p o f

is

uniformly

G/K H.

to

G/H

Thus from

obtain Let

G

(resp. G/K

be a t o p o l o g i c a l

Suppose t h a t

complete) is

G/H

g r o u p and

let

and

are

H/K

in the right quotient

totally

bounded

(resp.

uniform

complete)

in

structure.

In c a s e

Η

is defined,

the

= G .

where

Then

Ν

the

the s u b s e t s

for

W of

be s u b g r o u p s .

t o t a l l y bounded

is

G/H

The c o n d i t i o n

The c o n d i t i o n

W.S.Η

and

this

with

properties

be t r a n s l a t e d

i n t e r s e c t i o n of

Since the natural open i t

of

invariant general

The c o n d i t i o n

such t h a t

classes

translations

t h e u n i o n of

W £ W.

G

G,

to the r i g h t quotient

spaces.

H.

all

of

homo-

r e l a t i o n on

the natural p r o j e c t i o n

course,

these quotient is

group

G

equivalences.

spaces can,

separated

are continuous

the t o p o l o g i c a l

(left)

G

but in general

are the equivalence

The v a r i o u s u n i f o r m l y

form f o r

G

continuous,

with respect

are uniform

uniform

G χ G

Since the equivalence

structure,

open,

structure. of

uniformly

be a subgroup of

r i g h t uniform for which

function

function

is

a normal subgroup of

a s a toDoloqical a r o u p

G

the f a c t o r

group

( t h e r e i s no n e e d

to

. James distinguish

between

G/H

and

H\G

natural projection

n : G

therefore

continuous.

uniformly

Note t h a t G

and

G'

if

G/H

φ : G + G'

are t o p o l o g i c a l

is

groups,

equivalence

where if

and o n l y i f

The s e m i d i r e c t of

Η

is

product provides

g i v e n by a group

morphisms. the s e t and

Η X Κ £ Κ

Then

K,

topological

φ

a

topological

is a local

uniform

a good i l l u s t r a t i o n At t h e

algebraic

product

Κ

Η f< Κ

of

some

level

through

i s defined

autoto

h,h'

be

c Η

= ( h . h ' , a ( h ' , k ) .k·) Κ

denotes

are t o p o l o g i c a l

Η

is

open.

by

σ : Η χ Κ Κ

G"

w i t h the group o p e r a t i o n g i v e n for

( h , k ) . (h· , k ' )

and

e p i m o r p h i s m , where

i s uniformly

a c t i n g on a g r o u p

The s e m i d i r e c t

k,k·

where

Η

homomorphism,

discrete.

t h e p o i n t s we h a v e b e e n ma)cing.

this

φ

Moreover is

The

continuous

then

: G/H

Η = )ter φ .

case).

a continuous

i s an o p e n

Hence t h e i n d u c e d f u n c t i o n isomorphism,

in t h i s

the a c t i o n .

g r o u p s and t h a t

with the product group,

topology,

moreover the f i r s t

Now s u p p o s e t h a t σ

is

Η

continuous.

has t h e s t r u c t u r e of

projection

Η κ κ

Η

a is

open. The c o n s t r u c t i o n external

semidirect

semidirect algebraic

level,

let

Η

to d i s t i n g u i s h

G.

(external)

and a homomorphism

φ(h,k)

it

as f o l l o w s .

from t h e Returning

b e a s u b g r o u p and l e t

a g i v e n group

so that the

defined,

product

product which a r i s e s

subgroup of tion,

we h a v e j u s t d e s c r i b e d may be c a l l e d

Then

Κ

a c t s on

Κ

semi d i r e c t

product

Η >< Κ

φ : Η Κ Κ

G

hk

The c o n d i t i o n

for

Φ

t o be i n j e c t i v e

the condition

for

Φ

t o be s u r j e c t i v e

internal to

is given

by

conjugais

by

(heH,k£K). is

that is

that

the

be a normal

Η

=

the

Η η Κ =

(e);

H.K = G.

Topological groups When b o t h t h e s e c o n d i t i o n s and we r e g a r d and

K.

β : Κ

G

Note t h a t H\G

is

G

continuous,

the pair

open.

case

In t h a t

and a u n i f o r m

G

is

and

Η

K.

continuous.

isomorphism,

if

since G.

U

Κ Κ

consisting

of

The a c t i o n of

φ

K.

is

defined

are

acceptable

if

e m b e d d i n g when

is

Η

G/K

is (H,K)

is

topological

"'"^(αυ)

the equivalence

(e)

isomorphism

the pair

then

also

Urider

a topological

and h e n c e a

case.

Μ η Κ =

in addition.

that

and

i n any Φ

a : Η

II

as a t o p o -

i s continuous,

the semidirect

G

is

pairs

(x,0)

(x,y)

β

= U.K

w i l l be d i s -

Next l e t

pairs

(H,K)

isomorphic

(x,y)

(l,y)

quotient

condition

colleagues,

was t h o r o u g h l y

while

Κ

(I,χ

in this Η * Κ ,

subgroup is

the

χ = 1.

^y) .

It

case, moreover

spaces to right

was i n t r o d u c e d .

investigated

and we a d o p t t h e i r

the

Η

G/K.

who u s e d t h e t e r m " p r e s q u e i n v a r i a n t " the c o n d i t i o n

is

such t h a t

into

isomorphic to to

Η

y = 0

are a c c e p t a b l e ,

u s t u r n from l e f t

The f o l l o w i n g

i s p r o v i d e d by t h e

Here

such t h a t

transforms

topologically

topologically

product

considered e a r l i e r .

easy to see that

spaces.

Η Κ Κ

open in

normal subgroup c o n s i s t i n g of

is

and

action

open,

of

Η

bijection

s e n s e and t h e

assuming

is

and a

Η

later.

G =

and s o

of

H.K = G

is

The s t a t u s

An e x a m p l e o f

is

G/K

The i s o m o r p h i s m

«

isomorphism

Then

we o b t a i n

Still

an

group.

a uniform

therefore,

that

group

(H,K)

Φ

a c c e p t a b l e we f i n d

cussed

so t h a t

e q u i v a l e n c e when

these conditions,

IS o p e n i n

: Η

a topological

The homomorphism

Let us say t h a t

obviously

α

in the t o p o l o g i c a l

l o g i c a l group.

between

is

is

s e m i d i r e c t product of

an i s o m o r p h i s m

that

are subgroups

Η " Κ + Κ

(internal)

Φ

a r e g i v e n by t h e i n c l u s i o n s

Now s u p p o s e Κ

as the

are s a t i s f i e d

term

quotient

by P o n c e t

Subsequently,

[20^

horfever ,

by R o e l c k e [ 2 1 ] and

"neutral"

instead.

his

. . James Definition

(5.9).

is

in

neutral

there e x i s t s V.H c H.U

The s u b g r o u p

G

if

for

For

(5.10).

let

U

for

of

G

e

such

group in

G

G

that

may be s u b s t i t u t e d

Let

totally

condition,

Η

G.

Then

Η

b o u n d e d we h a v e of

is

g^W^ c V . g ^

.

S.W c V . S

By ( i v )

of

for of

Consider and

the

in

in

(5.1)

G .

G .

Then Since

of

e

intersection

Η

subset

there e x i s t s ,

W^

of

G.

some f i n i t e

a neighbourhood

normal

bounded s u b g r o u p

e

e

for

subgroups.

neutral in

of

Η c v.S

Η .

g^ £ S ,

V

Obviously

a s do open

be a t o t a l l y

some n e i g h b o u r h o o d

each element

in

for

G

such

W = W^^ π . . .

η W^ .

so

H.W c V . S . W <= V . V . S

c u.S

required. Subgroups of

particular G X G

if

and

is

is

is

(5.11).

For l e t

in U

for

neutral

SIN

group then so i s

neutral Let

G

G χ G .

in

G x G .

group such

is

an

b e any n e i g h b o u r h o o d

of

e

in

G.

SIN

of

e

in

G .

some n e i g h b o u r h o o d

V

in

(WxW).(AG)

G χ G

we h a v e

W of

e

(w,e).(g,g)

Therefore

g'

=

in

G .

= ( g ' , g ' ) . (v^,V2)

Then Since

^ (AG).(VxV)

for

that

group.

For each e l e m e n t

and s o

product

C o n v e r s e l y we h a v e

be a t o p o l o g i c a l Then

In

the d i r e c t

G

some n e i g h b o u r h o o d w e W we h a v e

groups are o b v i o u s l y n e u t r a l .

i s an AG

neutral

c υ

SIN

G

so

Proposition

AG

in

be any n e i g h b o u r h o o d

s =

AG

e

H.V c u . H

the stated

the t o p o l o g i c a l group

as

of

U

s i n c e b o t h s i d e s may be i n v e r t e d .

Proposition

We h a v e

V

the relation

subgroups s a t i s f y

that

the t o p o l o g i c a l

.

V.H c H.U

is

of

each neighbourhood

a neighbourhood

Of c o u r s e

V.V.c U

Η

some

for g € G

g'

w.g = g.v^^.v^

t G

and and

e g , V ^ .V,

Topological groups Thus

W.g C g . u

for a l l

g « G,

as

required.

F o r a n e x a m p l e of a s u b g r o u p w h i c h return to the semidirect product before.

If

a,b,x

ε B*

(a,0) . (x,y) . (b,0) Thus

G = H.v.H

neutral

and o n l y

Η

if

y ε Κ

V e N,

in

uniform

t o p o l o g y of

product

Η lx Κ

a uniformly

on

G . H\G

for

uniform

determined

continuous

11

that

function

the usual of

if

on

equiv-

G

for which t h e

if

satisfied

the

i s the

semidirect

Η tx Κ -»- Κ

Κ .

induces

Since t h i s

function

is

uni-

topology.

G

H\G

β

right

by t h e r i g h t q u o t i e n t

is

β : Κ + H\G

a uniform e q u i v a l e n c e

H.

is

in

G,

Also

the subsets

S c G; G/H

quotient

is

for

this

is

H\G

uniform

is

W.H,

i s uniformly

Il.li"

bounded i f f

d i t i o n as t h a t for For l e f t

of

the subsets

some f i n i t e

quotient

the s t r u c t u r e

the s t r u c t u r e

totally

so that

invariant properties G/H,

the i n t e r s e c t i o n

the union of

for

we s e e t h a t

is neutral

As i n t h e c a s e

H\G

not a

in the general c a s e ,

second p r o j e c t i o n

space with the right

translate

to

as

under

assumption. When

if

is

When t h e c o n d i t i o n i s

example,

then the

earlier

Η

i s compatible with the

inverse t o the uniformly continuous discussed

Η

that

therefore

structure coincides with the quotient It follows,

this

we have

G ,

are right c o s e t s

structure

with

G.

is neutral

uniform

κ Κ ,

we

.

the equivalence r e l a t i o n

alence c l a s s e s

form

G =

= (axb,ay)

whenever

s u b g r o u p of

Clearly

and

i s not neutral

not,

t o be t o t a l l y

structure,

into specific separated where

set

as a we may

terms.

if

and

W ε W,

conneced i f

the full

each

is defined

only

reduces

and o n l y

G.

Again

W e W we h a v e H.W.S in general, bounded,

s p a c e s t h e c o u n t e r p a r t of

if

= G

t h e same c o n -

namely (5.8)

W.S.Η = G. is

as

I.M. James follows . Proposition

(5.12)

Κ c Η c G Suppose

H\G

in the r i g h t

(resp.

topological

complete) consider G,

group

m*(F,G)

F.G,

is

H

is

are

(i) ,

conditions

for

topological

on

in the

the s e p a r a t e d

completion

t h e m i n i m a l Cauchy f i l t e r

on

G

generated

a r e m i n i m a l Cauchy on

t h a t of

a semigroup,

(iv)

is

for

so that a l l (ii),

G

SIN g r o u p s , G

(ii)

satisfied.

inverse

Cauchy f i l t e r let

for

for

Of c o u r s e

G

Thus

corresponding

function

but n o t ,

t h e c o m p l e t i o n of t h e new u n i f o r m

: G

which t h e l e f t

arising

from

the r i g h t uniform

general,

con-

inversion Cauchy

and t h e

condition

group,

structure,

the t o p o l o g i c a l structure,

s t r u c t u r e of

and

coincides with the

old.

the

group

arising G.

to

condi-

a homomorphism.

a new u n i f o r m

as the o l d uniform

structure

is

in

The c l a s s

the minimal F

Cauchy

Cauchy

Then an

G

a

our a t t e n t i o n

i s now a t o p o l o g i c a l

now a c q u i r e s

structure

as w e l l

i

G

of a r i g h t

us c o n f i n e

example.

G

the

inversion.

i n t o t h e m i n i m a l Cauchy f i l t e r

and t h e c a n o n i c a l

structure,

the

In f a c t

the

the

sense,

l a c l c i n g t o malte

i s g i v e n by t r a n s f o r m i n g

F

r i g h t uniform

Clearly

w i t h t h e r i g h t Cauchy c o n d i t i o n .

filter is

imply

concerning

By

in the a l g e b r a i c

that is

always a l e f t

G.

commutative.

observed,

topological

C

the

defined,

groups

function

of

totally

is

the c l a s s

all

G

is

G

However,

tains

K\G

complete)

m* : G χ G

G,

tion coincides

(resp.

Then

structure.

a r i g h t Cauchy f i l t e r . of

bounded

G.

function

(iii) ,

is

in

let

structure.

group i s

G

totally

neutral

same

As we h a v e a l r e a d y filter

Κ

structure.

commutative whenever G

conditions

and

g r o u p and

w i t h t h e r i g h t uniform

F.G

where

multiplication and i s

with K\li

a multiplication

where by

and

be a t o p o l o g i c a l

q u o t i e n t uniform

Finally,

(5.4)

G

be subgroups,

that

bounded

Let

from

However,

Topological groups For l e t

H*

the neutral uniform

denote the neighbourhood f i l t e r

element

Ν

of

neighbourhoods

s y m m e t r i c members o f

G.

Then

D*[W],

where

N.

If

v.v"^

W*

is

in

G*

g e n e r a t e d by the

W runs through

c w,

of

where

the

V e W,

then

V* c D*[W] c W* , where

W* d e n o t e s

contain

W

therefore, of

pairs

tnat

the s e t

o f m i n i m a l Cauchy f i l t e r s

a s a member. is

The new u n i f o r m

g e n e r a t e d by t h e

G.F

^ e W* ,

while

on If

G F

such that

and

Μ-Μ"-*· c w,

verse, then

let

in the

V €

be

o l d uniform

i.e.

Μ

c V.

structures So f o r

G

V =

V.M ε F

Therefore

filters

and

then

i.e. structure

To p r o v e t h e V . V . V t, w. Ν e G loss

and

of

V.M ε G,

If

conG.F"'",

Μ e F.

generality, and we have

t h e o l d and t h e new u n i f o r m

asserted. class

the s t r u c t u r e

function

i n comirion

new u n i f o r m

and

of

topological

and r i g h t Cauchy f i l t e r s

c a n be g i v e n

the canonical

in the

real

as

consistinc

such

W £ G.F~^,

i s D.^-small , w i t h o u t

the special

left

Μ

some

coincide,

G

G,

structure

for

We h a v e

G χ G

structure.

c ν

V.M.M ^.V c V . V . V c W.

which the

that

and s o

We may s u p p o s e

on

of minimal Cauchy

and s o

an e n t o u r a g e

such

which

h a v e a D ^ - s m a l l common member.

e G.F"^

V ε Ν

on

h a v e a D^^^-small member

Therefore

an e n t o u r a g e

(F,G)

consisting

F,G

hence

G.F ^ € W*. is

G

of

t h e o l d uniform

i s g e n e r a t e d by t h e s u b s e t s (F,G)

structure

subsets

o f m i n i m a l Cauchy f i l t e r s

F

i

: G

of

groups

coincide,

a topological G

G,

the

group,

for

completion so

i s a homomorphism.

that We r e f e r

Λ

to

G

that

it

(when d e f i n e d )

as t h e group c o m p l e t i o n of

has the following

Condition

(5.13).

where

is

Η

Let

a Hausdorff

characteristic

φ : G ->- Η

and n o t e

property.

be a c o n t i n u o u s

topological

G

group.

homomorphism,

Then t h e r e

exists

oo

i.m.jumes

one and o n l y o n e c o n t i n u o u s

homomorphism

ψ : G -·• Η

such

that

ψ1 = φ. In f a c t function,

(4.27)

and i s

s i m p l y go back it

obvious.

shows t h a t

unique.

exists

ψ

To show t h a t

to the d e f i n i t i o n

of

φ,

as a u n i f o r m l y ψ

is

in

(4.21),

continuous

a homomorphism we w h i c h makes

6.

Uniform transformation

By a G - s e t ,

for

groups

a g i v e n group

G ,

we mean a s e t

t o g e t h e r w i t h an a c t i o n of

G

ing

Under t h e a c t i o n t h e e l e m e n t

the usual conditions.

transforms

χ e X

thus defined p^ : G X .

into

is called

X

x.g

on t h e r i g h t

e X .

g

into

The t r a n s l a t i o n f u n c t i o n s In t h i s

u a t i o n where functions In t h i s

where

X

is

a uniform

g

·

The

are b i j e c t i v e ,

We s h a l l

is uniformly

of

s p a c e and e a c h o f

G

hence

X

of

the c o n d i t i o n

which i t

is

In f a c t

the

is

follows

independent

therefore

of

sit-

translation equlvalenci

t h r o u g h uniform

the

with r e s p e c t it

at

course.

i s a uniform

a c t s on

continuous,

structure.

structure,

: X ->- X

evaluation

a l s o be c o n c e r n e d w i t h t h e

r i g h t uniform that

g^

g £ G

function

is called

i s a t o p o l o g i c a l g r o u p and e a c h o f

functions

(5.1)

x.g

continuous,

s i t u a t i o n we s a y t h a t

G

by

satisfy-

s e c t i o n we s h a l l m a i n l y be c o n c e r n e d w i t h t h e

i s uniformly

equivalences.

X ,

The f u n c t i o n

translation

which sends

of

X

situation

evaluation to

left

or

a t o n c e from the choice

unnecessary to specify

(iv)

of in

this

connection. As A r e n s

[ 1 ] has demonstrated,

important i n the uniform is

i n the t o p o l o g i c a l

t h e o r y of

theory.

equicontinuity transformation

Looking f i r s t

is

as

groups as

it

a t the family

of

t r a n s l a t i o n s we make Definition soace

X .

(6.1).

Let

G

be a g r o u p a c t i n g

The a c t i o n i s u n i f o r m l y

on t h e u n i f o r m

equicontinuous

if

for

each

yu

im.jumti

entourage that

D

of

(Cg,ng)

X

e D

For e x a m p l e , lation.

left

structure.

uniform When

whenever

suppose

The a c t i o n

uniform

there exists

is

that

D*

of

(ζ·9,η·9)

G

X X X foi·

equicontinuous

if

and o n l y

for

consisting some

the uniform If

uniform

G

spaces

and

X

any e q u i v a r i a n t

of

Y ,

and u n i f o r m l y

open i f

G

on

X

is

the condition

D

of

X

space

X

neutral

if

entourage

(6.2) .

is

(ζ,η)

such

the a c t i o n

is

uniformly

,

where

D*

that

an i n v a r i a n t

i s uniformly it

D

base

is

on t h e

on X ,

continuous

then if

it

Ε

h £ G ,

c l a s s e s are the o r b i t s condition

structure,

G

than i s

relation

χ.Π

for

in the sense

(χ t X)

R of

t o be

Section

n e c e s s a r y and s o we

be a g r o u p a c t i n g

equivalences.

each entourage of

X

such that

implies that

The s t a t e d d i t i o n of

Let

(2.14)

condition in t h i s

D

of

X

(ζ,η.h)

(C.g,n)

£ D

is precisely special case.

is

open.

d e t e r m i n e s an e q u i v a l e n c e

stronger

through uniform for

the

a sub-

introduce

Definition

and

in

right

through uniform

transitively

a sufficient

compatible w i t h the uniform

also

in the

equicontinuously

and a c t s

X » Y

equicontinuity

However,

X

forms

a c t s uniformly

oi: w h i c h t h e e q u i v a l e n c e

Uniform

trans-

an SIN g r o u p .

entourages

X ,

by r i g h t

equicontinuous

is

of p a i r s

function

The a c t i o n of η

space

such

structure.

the group

continuous

G

If

X

equicontinuous

if

g e G .

then the family

of

g ε G .

each entourage

r u n s t h r o u g h t h e e n t o u r a g e s of for

and

Ε

a c t s on i t s e l f

a c t s on t h e u n i f o r m

ε D

£ Ε

The a c t i o n i s u n i f o r m l y

e q u i v a l e n c e s we c a n d e f i n e set

(ζ,η)

always uniformly

structure

G

an e n t o u r a g e

on

the

The a c t i o n there e Ε , for

uniform is

exists

an

where

ζ,η

sane

g € G .

the ccmpatibility It

implies

£ X

that

conthe

2.

.

Uniform transformation groups natural projection q u o t i e n t unifonn formly

π : X -»• X/G

structure,

continuous,

right

i n v e r s e of

π

For e x a m p l e , i c a l group of

suppose

G

l e f t uniform

structure.

i n t h e s e n s e of

t h e a c t i o n of

G

on i t s e l f

if

(5.9).

G

topolog-

The

action

and o n l y In

i s always n e u t r a l ,

equicontinuous unless

co-

continuous

i s a s u b g r o u p of a

in

Γ ,

x/G

section.

that

with the

on

A uniformly

neutral

formly

Γ

is called a

as w e l l as uni-

topology

translation is neutral

is

on

the uniform

topology.

s p a c e , w i t h the

open,

by r i g h t

G

G

Γ ,

i s uniformly

and t h a t

incides with the quotient

to the o r b i t

if

particular

but i s not u n i -

admits s m a l l

invariant

neigh-

is

we d e s c r i b e

bourhoods. When t h e c o n d i t i o n

in

as a neutral G-space. are a l s o n e u t r a l , family

of

(6.2)

Invariant

satisfied

s u b s p a c e s of

a s can e a s i l y be s e e n .

Also if

is

also a neutral

i s uniformly

are t o t a l l y bounded, and a r e u n i f o r m l y connected.

uniform then

is

(2.17)

If

i s a uniform

implies

l e t us suppose

Then t h e o r b i t s G

is

totally

whenever

each p o i n t

G

that

G

of

that G

bounded,

is

χ ε X ,

χ

and

H\G

uniformly

induces

has the l e f t

i s uniformly

ecjuivalence whenever

and X

of

the action

Moreover

bounded w h e n e v e r

for

with

bijection

the s t a b i l i z e r

equivalence.

(1.10),

IIX^ ,

a

,

structure.

σ^

whenever

by

p^ ,

continuous

σ^ : H\G - xG Η

(1.9),

connected,

Moreover

a uniformly

where

by

continuous.

is

G-space.

T u r n i n g now t o t h e e v a l u a t i o n f u n c t i o n s these

G-spaces

{X^}

n e u t r a l G - s p a c e s then the uniform product

the diagonal a c t i o n ,

e a c h of

neutral

X

(2.16) X/G

implies that are t o t a l l y

i s uniformly

X

qiiotieni

equicontinuous is

a

is

totally

bounded,

topnl©glral

vhilo

c o n n e c t e d whenever

G

.jnd

I.M. James X/G

are uniformly

Definition

connected.

(6.3) .

Let

G

be a t o p o i o g i c a l g r o u p a c t i n g

uniform

space

X

uniform

if

each entourage

for

bourhood g £ U

U

of

and

χ

through uniform e q u i v a l e n c e s .

e

in

uniform

such t h a t

i s uniformly structure.

same c o n d i t i o n a s u n i f o r m r i g h t uniform same i f

G

or c c m p a c t ,

X

The a c t i o n

there e x i s t s

(x.g,x)

the action i s uniform

evaluation functions left

of

the

« D

a

is

neigh-

for

all

ε X .

In other words,

to the

G

D

on

i s a n SIN or i f

as a uniform G-space.

in

not,

(6.3)

with respect

if

respect the

to

are

the the

G

is

abelian

is discrete

or

trivial.

satisfied

continuous

of

in g e n e r a l ,

the conditions

G is

with

we d e s c r i b e

Uniform G - s p a c e s form a c a t e g o r y

the equivariant uniformly of

is

in particular

t h e t o p o l o g y of

When t h e c o n d i t i o n

The e q u i v a l e n c e s

This

Of c o u r s e

group,

the family

equicontinuous,

equicontinuity

structure.

if

functions

the category are c a l l e d

X with

as morphisms.

equivariant

uniform

equivalences. Invariant as can e a s i l y G-spaces action,

s u b s p a c e s of u n i f o r m G - s p a c e s a r e a l s o be s e e n .

then t h e uniform is

of u n i f o r m

the

of

a base for

ΠΧ^ ,

consider

t h e uniform

convergence.

is

a family with the

of

uniform

diagonal

G-space.

illustration,

self-equivalences

tXj}

product

a l s o a uniform

By way o f

since

Also if

uniform,

the group

space

X ,

The a c t i o n o f

t h e neighbourhoods of

G

G

of

with the on

X

is

uniform topology uniform

the identity c o n s i s t s

subsets {φ £ G :

where

D

example,

runs

(φ ( x ) , x )

£ D

V

X

£ X}

through the entourages

consider

the group

G

of

of

, X .

biiections

For

another

θ : Τ + Τ ,

of

Uniform transformation groups where

Τ

is

us give

G

an i n f i n i t e

{Θ e G : θ ( t ) S

not

Proposition

= t

X X G ->• X

(6.4).

Let

structure D

on

i s uniformly

X

subsets

subsets

of

Τ ,

and s o

the

such that

(ζ.h,η.h)

such t h a t

If

then the a c t i o n with respect

the

function

to the

left

G .

(n.h,n.k)

h~^.k e U .

D'

of

X ,

for

all

Then i f

for

(ζ,η)

be a

Since the

£ Ε

and

action

Ε

of

h £ G .

a neighbourhood

all

(ζ,η)

D'

an e n t o u r a g e

there e x i s t s £ D'

and l e t

» D' c D .

there e x i s t s

£ D'

the a c t i o n i s uniform

such t h a t

the

for

,

continuous,

ecpaicontinuous

such t h a t

e

of

let

A base

be a u n i f o r m G - s p a c e .

be a n y e n t o u r a g e

symmetric entourage

of

£ S}

as d i s c r e t e

convergence.

consists

equicontinuous,

i s uniformly

For l e t

Since

V t

Τ

unifonn.

action i s uniformly

X

the i d e n t i t y

runs through the f i n i t e

action is

uniform

Regarding

t h e t o p o l o g y of p o i n t w i s e

t n e n e i g h b o u r h o o d s of

where

set.

η £ X

£ Ε

and

and

U

li,k £ G

(h,k)

£ L·^

we h a v e

(ζ.h,η.h) and s o

ε D',

(ζ.h,η.k)

Of c o u r s e

in

(6.1)

By way of where

G

is

ε D ,

as

of

ε D'

required.

t h e c o n v e r s e of

uniform c o n t i n u i t y dition

(η.h,η.k)

(6.4)

holds,

the action function

and t h e c o n d i t i o n i n illustration

a topological

in the sense

implies

that

both the

(6.3).

s u p p o s e t h a t we h a v e a G - s e t group.

con-

Consider

the family

X , of

subsets Ly = { (x . g , x ) of

X X X ,

where

: X ε X , g ε U) U

r u n s t h r o u g h t h e n e i g h b o u r h o o d s of

e

yn in

ι.m. James G .

We r e f e r

by t h i s acts

family

G

structure

a s t h e natur^_l u n i f o r m

trivially

When

t o the uniform

the n a t u r a l uniform

acts

transitively

on

X

generated

structure.

structure

is

t h e n a t u r a l uniform

When

G

discrete. structure

is

trivial. Returning

to the general

n a t u r a l uniform equivalences, Moreover,

structure.

i s uniformly Recall

Proposition

continuous.

When

Let

is

sense.

X

of

Let

such t h a t

X

is

{U^ : X e X} say. of

D

that

X ,

V^

of

action

X

e

χ

of

Ό .

G

the closed

n-ball

0(n)-spaces,

i n the standard wav.

(6.5), B'^

X

Since U^

such t h a t

G-space,

with the

of

unique

in

the

the

admits a

neigh-

the a c t i o n

V

r, of

χ

in

is X

covering

is a

X]^ , . . . ,χ^^ , neighbourhood

(6.3). be s e e n

directly,

(n-1)-sphere

s""^

are

where t h e o r t h o g o n a l group

0(n)

acts

Hence t h e o o e n n - b a l l

and

U^.V^^ <= W^ .

i n d e x e d by

η V

and c a n e a s i l y and t h e

of

from t h e o p e n

subcovering

the requirements

from

neigh-

X

in

a finite

It follows

admits

are the

be any n e i g h b o u r h o o d

c o m p a c t we c a n e x t r a c t of

space

we u s e t h e term G - s p a c e

t h e r e e x i s t s a neighbourhood

satisfying

uniform

Then

W X W

Then t h e i n t e r s e c t i o n e

well

G-space.

X

a neighbourhood Since

as

group the

be a c o m p a c t r e g u l a r

(6.5)

X

topological

open,

a compact r e g u l a r

X χ X. Then e a c h p o i n t

W

uniform.

We p r o v e

a uniform

In the h y p o t h e s i s

bourhood

i s an SIN

is a topological group.

in

obviously

in which the entourages

the diagonal.

iniformization,

diagonal

through uniform

are uniformly

G

the

equicontinuous.

(6.5).

topological

is

be a G - s e t w i t h

and t h e a c t i o n i s

from S e c t i o n 3 t h a t

bourhoods of

X

The a c t i o n

i n any c a s e ,

a unique uniformization,

G

let

the evaluation functions

as uniformly

where

case,

b'^ - s " ^

is

Uniform transformation groups also

a uniform

0(n)-space,

being

However t h e E u c l i d e a n s p a c e for

It"

is

of

G ,

which i s

Definition l e n c e s of

important

(6.6).

Let

t h e uniform

s u c h Jihat

(ζ.<3,ξ.)

For e x a m p l e , ical

group e

left

let

if

of

B" .

0(n)-space,

G

involving

the

topology

applications.

be a g r o u p o f u n i f o r m X .

The a c t i o n i s

there e x i s t s

for

all

U π G = {e}

The c o r r e s p o n d i n g structure s a t i s f i e s

self-equivauniformly

an e n t o u r a g e

ζ c X

be a d i s c r e t e

Then

Γ .

uniform

G

/ D

Γ .

in

right

and

g e G -

subgroup of for

D

of

χ

{e}.

the

topolog-

some n e i g h b o u r h o o d

entourage

Ry

of

the requirements

Γ

for

ϋ

in the

the action

translation.

Clearly is

not a uniform

not

in the

space

properly discontinuous

of

subspace

η 2 2 . There i s another c o n d i t i o n ,

of

an i n v a r i a n t

properly

a properly discontinuous action discontinuous

in the uniform

in the topological

sense.

sense

In t h e

other

d i r e c t i o n we h a v e

Proposition lences

of

(6.7).

Let

For s u p p o s e the t o p o l o g i c a l

U^

If

sense.

g £ G - (e)

entourage

D

of

meets

implies

that

continuous

If

i n t h e uniform

the action

is

sense then the

action

sense.

Then t h e r e e x i s t s U^

.

X

By

e D D[g ] g = e

such that (3.19)

such t h a t

we now h a v e

Dlζ].g

X .

self-equiva-

that the a c t i o n i s properly discontinuous

a neighbourhood

for

space

in the t o p o l o g i c a l

properly discontinuous

X £ X

b e a g r o u p of u n i f o r m

t h e compact r e g u l a r

properly discontinuous is

G

and, .

for

tor

Therefore

i n t h e uniform

sense.

U^.g

each

all

ξ,

point

d o e s not meet

there e x i s t s a

some

since

for

in

symmetric

Οϋξ] c u^

ξ e X

Οίζ ] c υ,^ ,

and

g a (J

for

some

then

this

the action i s properly

dis-

χ

yo

i.m.jumes Properly

discontinuous

actions.

When

is

by

defined

τ (x,x.g)

G

acts

= g

where

R

under

π x ττ .

is

function.

Proposition

(6.8).

structure

equicontinuous section

of

G .

This follows mutually

of

X

a function

τ

the diagonal

of

τ

is

be a f r e e

of

free

: R

G

X

called

G-space with to

the action

admits a uniformly

Then

X/G

scmetimes

with respect

Suppose t h a t

and t h a t

product

X

cases

,

function,

Ο •. χ / G '* X .

the uniform

X

literature

Let

translation

are s p e c i a l

on

the preimage

i n the

translation

form

freely

(xeX.giG)

X x X

continuous

actions

i s uniformly

the

uniformly

the

left

is

uni-

uniformly

continuous equivalent

to

X/G * G .

from t h e o b s e r v a t i o n

inverse uniformly

continuous

that

there

exist

functions

X ^ X/G X G ^ X , where

θ(χ)

(X£X,giG) If

=

(π ( χ ) , τ (σπ ( χ ) , x ) )

the

(sepaurated)

topological

group

equivalences

ccmpletion

Specifically just

φ(x.G,g)

=

a(x.G).g

.

through uniform

is

and

the

X

of

acts

then

of

translation

the ccmpletion

G

X of

G

on t h e u n i f o r m also acts

through uniform X

on

space

the

equivalences.

by a g i v e n e l e m e n t

the corresponding

X

translation

g ε G of

X .

We p r o v e Proposition uniform X

of

(6.9) .

space X .

X .

Suppose t h a t Then

G

G

acts uniformly

acts uniformly

on t h e

on

the

completion

Unifo

transformation groups

I n o t h e r words t h e c a n p l e t i o n of a l s o a uniform G - s p a c e . t o show t h a t f o r

To

a uniform G - s p a c e

establish

this

each symmetric entourage

it

D

is

of

sufficient X

there

e x i s t s a neighbourhood

U

of

e

with the following

for

e a c h Cauchy f i l t e r

F

on

X

and e a c h e l e m e n t

the

induced f i l t e r

with

(c^) jjF

such t h a t contains

g e U

a D ' - s m a l l member

and

(ς·9ίη)

U

of

e

χ e X .

ε D' ο D*

(i.g,n.g)

in

of

U

canmon

ε D'

the

Proposition

(6.10).

· D'

of

.

By ( 6 . 3 ) (x.g,x)

and

ζ,η

ε D* « D' Hence

c^M

υΜ

for

then

and

so

Μ "Μ

is

i s canmon t o

F

there exists

e D'

ε Μ

X

a

all (ζ,ηΐ

c D" ,

D'«D'·0'-small F

and

result.

is discrete.

Let

If

X

be a u n i f o r m G - s p a c e ,

t h e a c t i o n of

properly discontinuous of

say.

g « U

Since

t h i s proves

M,

(i,n.g)

· D'

D'

By t h e Cauchy c o n d i t i o n

such that

If

,

and s o υ - s m a i l .

G

then so i s

on

X

is

where

G

uniformly

t h e a c t i o n on t h e

completion

X . To e s t a b l i s h t h i s

it

is

sufficient

t o show t h a t

an e n t o u r a g e

D

with the property that if

X

F

and

(c^)*F

then

g = e

such that some

g ε G

member,

where

b o t h members o f any

g

c h o o s e a symmetric e n t o u r a g e

D ' s D ' o D ' c D .

neighbourhood

for

property:

F . To s e e t h i s f i r s t

X

has a D - s m a l l manber

is

ζ,ε Μ η

(ζ.^,ί)

D

i s as in F

admits

is a filter

we h a v e

on

h a v e a D - s m a l l member i n canmon .

So l e t

(6.6).

Μ

Then

be a D - s m a l l common Μ

and

a^®

and s o h a v e n o n - e m p t y i n t e r s e c t i o n .

e Μχ Μ C D .

uniformly

F

X

ξ ε Μ

and

ζ.g

S i n c e t h e a c t i o n of

p r o p e r l y d i s c o n t i n u o u s we o b t a i n

ε Μ , G

therefore

on

g = e

For

Χ ,

is as

required.

7.

Uniform spaces over a base

In t h i s

s e c t i o n we work o v e r a u n i f o r m b a s e s p a c e

By a u n i f o r m

space over

with a uniformly projection. Β

Β

continuous

Usually

X

identity

function

alone

c a n a l w a y s be r e g a r d e d

the

I mean a u n i f o r m

as p r o j e c t i o n .

T,

using

the

first

projection.

group a c t i n g uniformly

form s p a c e

X

as a uniform

form s t r u c t u r e , If

X

Again,

is

using

X/G,

the natural

projection.

a uniform

over

Β

by

and may be r e g a r d e d a s a u n i f o r m

say,

of

the p r o j e c t i o n .

the preimage i s Uniform

spaces

n o t i o n o f morphisra. with projections continuous function

called

over

Β

Let

When

of

B'

Β

is

Y

be u n i f o r m

function

over

Β

I mean a u n i f o r m l y

such that

are c a l l e d uniform

qφ = ρ .

c a n be uni-

ρ

the

denoted B'

by b,

X^^ . following

spaces over

Β

continuous

The e q u i v a l e n c e s

equivalences

continuous

uni-

By a u n i f o r m l y

over

are d e f i n e d i n t h e o b v i o u s way.

be a u n i f o r m l y

X

form a c a t e g o r y w i t h t h e and

a

on t h e

t h e f i b r e and d e n o t e d by

X

is

a point"space

respectively.

Pull-backs

G

space over

q

the category

if

is usually

and

Y

any u n i f o r m

then

ρ

φ : X

λ : Β' ->• Β

B'

Β χ Τ

with projection

ρ ^(B')

restriction

a subspace

with

with the quotient

preimage ΧβIί

of

space

itself,

equicontinuously

space over

Thus

product

for

through uniform e q u i v a l e n c e s ,

the

notation.

the uniform Β

together

called

space over

space over

topological

regarded

sufficient

Also

X

ρ : X -»· Β,

as a uniform

can b e r e g a r d e d a s a u n i f o r m space

is

space

B.

function,

of

Β Thus

let

where

B'

is

a

u mjorm spaces over a oase iniform

space.

form s p a c e space of (b',x)

Then t o e a c h u n i f o r m

λ*Χ

over

the uniform such t h a t

g i v e n by

B'

p'(b',x)

to

B' If

of

X

X

is

= b'

.

A

is

Β

is of

B'

c Β

with

a

uni-

the subpairs p'

λ

with the r e s t r i c t i o n

is

the

in-

X^,

of

space over

Β

and

A

X

if

Note

fibrewise

Xj^

is

that if

Β

is

λ*Χ,

a subset

by

restricting

We d e s c r i b e

the c l o s u r e

A

dense in

is

of

fibrewise where

A

Aj^

for

d e n s e in

λ : Β'

Β

X is

above. {Xj }

fibrewise

the uniform

with

is

uniform

coordinates

a family

of uniform

product

Π^Χ^

product

lie

JIX^

spaces over

i s defined

consisting

i n t h e Scime f i b r e .

of

Β

t o be the points

Thus

the

Π^Χ^

subspace

all

of

comes

whose

equipped

projections ..

: Π3Χ.

X.

which are u n i f o r m l y

continuous

the pull-back

of

λ*Χ

respect to a uniformly identified

with

space over

Β

B'

where

special

base which are important

over X

is

B.

over

function B'

Note Β

λ : B'

that

with Β

can be

r e g a r d e d a s a uniform

λ. classes

of uniform

i n our t h e o r y .

i s uniformly

which the p r o j e c t i o n In [ l l J

space

continuous

with projection

which the p r o j e c t i o n

functions

a uniform

>4„ X,

There are s e v e r a l

1.

consisting

and t h e p r o j e c t i o n .

dense^ i n b e B.

λ*Α

If

for

λ*X

as a uniform space o v e r

structure

each point

of

where

over

and t h e p r o j e c t i o n

In c a s e λ*Χ

a uniform

we r e g a r d

as fibrewise

as

χ X

X

.

the uniform

then

B'

= p(x)

c l u s i o n we may i d e n t i f y X

is defined,

product

\(b')

space

open.

One i s Another

i s a l o c a l uniform

d e n s e a l w a y s means f i b r e w i s e

spaces over a the c l a s s i s the

equivalence.

dense.

for

class Note

I.M.James that

if

the uniform

space

so does the pull-back Β

and u n i f o m i l y

Proposition

D

of

X

p"^(E[b])

is

ζ

B'

X

open i f

C

all

c ρΟ[ς]

for

all

to

loss

Then i f

and s o

and t h e n

ξ £ DtxJ

for

over a base.

χ

£ D[Xg]

.

so that

over

X

have the o r i g i n a l

.

choose

that

So i f

β £ E[b]

η £ 0[ζ]

χ χ

.

Then

as

property

property

if

in Section

Definition

(7.2). all

Β Also B.

is

for

1.

For

The u n i f o r m the fibres

of

for

is

that

seme ζ £ X^ ,

an o b v i o u s way spaces

t h e uniform

space

the fibres

of

procedure i s

Section

4 is

invariant

too

concerned properties

example space

X

X

separated.

always separated, Β X Τ

entourages

of uniform

this

the uniformly

the

for

is

all

Although

satisfied.

required.

there

we may s a y t h a t

is

we h a v e

£ Οίζ]

w h e r e t h e Cauchy t h e o r y o f satisfactory

e X^

,

spaces

fibrewise

property.

considered

sDace o v e r

that

so

condition

and

Hence

has the f i b r e w i s e

simplistic

itself.

D ,

seme

β £ pD[xJ

Specifically

X

Thus

entourage

Then we can

we may a s s u m e t h a t

β £ E[b]

a corresponding

if

B.

for each such

open.

the stated

Given a p r o p e r t y of uniform

is quite

Β

ξ e Xg

b = ρ(η)

of g e n e r a l i t y

b £ Εϋβ]

separated

of

symmetric and

suppose that

are symmetric.

it

We p r o v e

space over if

space

required.

Conversely,

Β

Β .

and o n l y Ε

then

e v e r y uniform

λ : Β'

i s uniformly

β £ Β

and s o

as

Without r e a l

ρ

for

property

b c Β .

in relation

b £ Ε[β]

£ 0[η],,

that

,

be a u n i f o r m

uniformly

for

has e i t h e r

function

an e n t o u r a g e

Ε ,

to define

Β

there e x i s t s

symmetric

then

Let

over over

continuous

For s u p p o s e

Ε[β]

λ*X

(7.1).

The p r o j e c t i o n

X

fibrewise

each separated

are

over

Β

as a uniform separated, uniform

is

space

fibrewise

over

as a uniform

space

T.

We p r o v e

u mjorm spaces over a oase Proposition

(7.3).

Suppose t h a t Then

X

Β

is

R

is

R

Since

X

is

X XgX

is

just

of

B. is

Since the just

the diagonal as

ΔΧ

space over fibrewise

Β .

separated.

t h e e n t o u r a g e s of the i n t e r s e c t i o n

latter

reduces

to the

product

hence

R

of diagonal

X x„X . ΰ

t h e i n t e r s e c t i o n of

ΛΧ,

X.

itself

R

with

reduces

to

required.

is closed

t h e s i t u a t i o n when Proposition

of

the f i b r e w i s e

separated

It is easy to see

uniform

is

i n t h e p r e i m a g e of

fibrewise

the diagonal,

uniformly

X

be t h e i n t e r s e c t i o n

the preimage

only if

be a uniform

s e p a r a t e d and

contained

the entourages ΔΒ,

is

X

separated.

For l e t Then

Let

101

that in

ΔΧ

(7.4).

X X x^X

is

Let

if

fibrewise

.

separated

Our n e x t r e s u l t

if

and

concerns

open. X

be a uniform

open p r o j e c t i o n .

equivalence

is

space over

Then t h e p r o j e c t i o n

and o n l y i f

the diagonal

D

is a

with

local

function

Δ : X Η- X XgX i s uniformly

open.

For s u p p o s e X

admits a base of

tne p r o j e c t i o n (D[x]

for

all

X

χ e X Δ

required.

If η

X £ X ,

D

is

(X XjjX)

equivalence.

Then

Δ

a basic

entourage

then

C ADCXJ

i s uniformly

i s uniformly

there e x i s t s

all

i s a l o c a l uniform

symmetric entourages which are transverse

and s o

(E[x] X E[x]) for

ρ

ρ .

X D[x])

suppose that of

that

open.

open.

If

a symmetric e n t o u r a g e

D

Conversely i s any

Ε

of

X

entourage such

η (X ΧβΧ) c Δ Ο [ χ ] and t h e n

Ε

is

transversa to

ρ ,

as

that

to

102

. James

Definition

(7.5).

wise t o t a l l y Thus

Β

The u n i f o r m

bounded i f is

always

space over i t s e l f .

space over

Again i f

is

formly wise

fibrewise

Proposition

X

over

of

X

totally

Β χ Τ B,

is

bounded, (7.6).

on a u n i f o r m

as a uniform Let

X

uniformly

open p r o j e c t i o n .

and t h a t

X

is

fibrewise

space

bounded.

as a u n i f o r m

totally

bounded,

X

then

X

X/G .

Β

bounded.

is

uni-

is

fibre-

We p r o v e

space over

that

T.

group a c t i n g

over

be a u n i f o r m

totally

fibre-

t o t a l l y bounded s p a c e

space

Suppose

is

bounded,

fibrewise

for each

Β

are t o t a l l y

a t o t a l l y bounded t o p o l o g i c a l

equicontinuously

totally

the f i b r e s

Also

as a uniform G

all

space

B,

with

totally

bounded

Then

X

is

totally

and l e t

Ε

be a s y m m e t r i c

for

b ε Β

and

subset

S

bounded. For l e t

D

b e any e n t o u r a g e o f

Β

such that

entourage

of

Since

Β

is

of

X

such t h a t

finite

number o f

totally

each fibre

is

subset

of

then

Τ

and s o Thus

fibres.

totally X

p(s) (5,t)

bounded,

for

some

= ρ(ς)

for

for

D c D[T] = X ,

(2.15),

some

uniformly

(7.7).

if

some

and s o

t ε T,

the union of

exists

.

ξ ε D[x],

a special

all

is

t o t a l l y bounded,

p~^pS c D [ T ]

The u n i f o r m

connected

ρ ^pS

s ε S,

all

a finite

and s o t h e r e

which proves

is essentially

Definition

Now

The u n i o n i s

such that

ε D

- pDLxJ

bounded t h e r e e x i s t s

Β = E[pS].

b e E[p(s)]

therefore

E[bJ

X,

ρ(s)

a

since

finite

So i f

x e

ε E[b] c pD[x],

hence

therefore (7.6).

a

χ ε Χ^^.

ξ ε ρ

^p(s)

χ ε D » D[TJ

Our e a r l i e r

.

result,

case. space

the fibres

X of

over X

are

Β

is

fibrewise

unifonnly

connected. Thus

Β

is

always fibrewise

self.

Also

Β >· Τ

uniform

space over

is B,

fibrewise for

uniformly

connected over

uniformly

connected,

each uniformly

connected

it-

as a

uniform

orm spaces over a ase space

Τ .

Again i f

G

group a c t i n g uniformly then

X

is

over

X/G .

fibrewise

is

equicontinuously uniformly

(7.8).

Let

X

uniformly

open p r o j e c t i o n .

connected

and t h a t

i s uniformly For l e t

X

is

on a u n i f o r m

connected,

be a uniform Suppose

space

as a uniform

fibrewise

space over

that

Β

is

uniformly

D

X

space

3,

with

uniformly

connected.

be any s y m m e t r i c e n t o u r a g e o f

b e B.

of

Let

Β

ζ,η

such

that

e X .

Then

connected.

chain to a point

ρ(ξ)

e Β ,

I assert

x^ £ Xj^

,

that

where

ζ

X,

and l e t

ECb] ^ pOLXj^J

Then

can be j o i n e d by an E - c h a i n uniformly

topological

connected.

be a s y m m e t r i c e n t o u r a g e each point

connected

We p r o v e

Proposition

X

a uniformly

and

since

Ε

for

ρ(η) Β

is

can be j o i n e d by a D-

i = l,...,n

.

This

is

i trivial true

for

for

some

(Xj^jX^^j^) ξ

1 = 1 ;

e D

make t h e

i < η . for

inductive

Then

hypothesis

£ E [ b ^ ] c poLxj^J

some

^ ^b.

'

n' X,

it

and

^^ r e q u i r e d .

can b e j o i n e d by a D - c h a i n t o some p o i n t

then since

that

η' £

is

so Thus

b

,

η

and

c a n be j o i n e d t o η by a D - c n a i n ( i n Xj^ i s uniformly connected. Thus ζ can be j o i n e d

to

η η

by a D - c h a i n a s r e q u i r e d .

(2.17),

is

essentially

a special

We s e e , t l i e r e f o r e , t h a t over a base properties point.

c a n be d e f i n e d of

This

uniform

t h e t h e o r y by

proposition uniformly

(7.9).

result,

case.

fibrewise

properties

which reduce

to the

of

uniform

a s we s h a l l

however, see.

is

spaces

corresponding

s p a c e s when t h e b a s e s p a c e r e d u c e s

simple procedure,

t h e Cauchy t h e o r y , of

Our e a r l i e r

inadequate

We b e g i n our

to a

for

discussion

proving Let

open p r o j e c t i o n .

X

be a u n i f o r m Let

F

space over

B,

be a Cauchy f i l t e r

with on

X

I.M. James such t h a t

p^F

member o f that

converges

trace

For

let

of

G

D

on

for

converges

we h a v e

to

b

and t h e n

Xj^ .

Since

G

runs

through

the

sufficiently

is

such

if

for

itself

X . Ε

By

of

Xj^ ,

for

so

F,

we

have

Since

p^F

sufficiently Thus

by t h e s e t s X

(7.1)

Β .

c D[Xj^] .

of

meets

D"^CM]

D ^[M],

and

Μ

runs

this

proves

small meets

where

through

0

the

the r e s u l t .

i s m i n i m a l Cauchy t h e n t h e

The u n i f o r m

each point converges

that

space

b ε Β to

b

this

reduces

X

over

In

trace

Β

is

of

F

fibrewise

and C a u c h y f i l t e r

,

F

itself

to completeness

Β

is

a point-space.

complete,

as

a uniform

fibrewise

complete

F

converges

on

to

X

some

uniform

in

Moreover

space spaces

over

the

Β

ordinary

is

itself.

always

fibre-

Pull-baclcs

over a base

are

also

of

fibre-

complete. We h a v e

at

once

Proposition

(7.11).

Let

space

Β .

Β

over When

uniform we

F

X ε Xj^ .

s e n s e when

wise

F

p,F

Note

wise

entourages

(7.10)·

that

point

if

of

pM c ECb]

generated

of

every

defined.

some e n t o u r a g e

Μ c p"^(E[b]) is

G

Then

defined,

Definition complete

is

s m a l l members o f

particular, X. b

Xj^

b e any e n t o u r a g e

ρ ^ E [ b ] <= DC Xj^J

on

b £ Β .

t h e m i n i m a l Cauchy f i l t e r

the

Μ £ F,

to a point

Β

is

If

X is

separated

space over

Β

be

a fibrewise

complete the

then so i s

fibres

are complete.

complete

of

uniform

X .

a fibrewise

In t h e o t h e r

complete direction

have

Proposition uniformly then

X

(7.12).

Let

open p r o j e c t i o n . is

fibrewise

X

be a uniform If

complete.

all

space over

the fibres

of

X

Β , are

with complete

umjorm spaces ver a os For l e t on

X

b

be a p o i n t o f

such t h a t

p^F

m i n i m a l Cauchy f i l t e r trace

is

G

since

itself

X^^

the r e s u l t . complete

X/R

of

G

Β

for

if

X

is a

F

(7.13).

formly

G

i s Cauchy.

X.

X

is

Then

X

is

uniform

2,

(7.14).

Let

G

be a complete

uniformly

equicontinuously

let

A

Let

be a f i b r e w i s e

fibrewise and l e t

Y

Then t h e r e e x i s t s ψ : X ->• Y

over

To d e f i n e procedure

ψ ,

as for

separated,

entourages

complete

over

is

uni-

χ ε X . group

space

Then

X.

acting X

X/G .

X

be a u n i f o r m

d e n s e s u b s e t of

space over

X .

such that

continuous

Y

Β

Y

and

be a

space over function

Β

over

continuous

B.

function

ψ|Α = φ .

as a f u n c t i o n , the ordinary

Let

complete uniform

be a u n i f o r m l y

S i n c e we a r e o n l y a s s u m i n g

the

topological

o n e and o n l y o n e u n i f o r m l y Β

of

each point

on t h e u n i f o r m

separated fibrewise φ : A

complete.

the projection

Corollary

(7.15).

Further-

space.

c o m p l e t e for

Proposition

Τ .

with projection a

fibrewise

is

complete over

proves

fibrewise

space Β

point

Hence

which is

be the i n t e r s e c t i o n

R[x]

fibrewise

,

fibrewise

also

is

χ

th

point

an a d h e r e n c e

Β χ Τ

space over

then

R

t o some

point since

that

of

The

is

converges to

filter Xj^

(7.9).

χ

as we h a v e s e e n i n S e c t i o n

open,

by

on

Then

uniform

the separated quotient For,

The t r a c e

e v e r y c o m p l e t e uniform

Let

space

be a Cauchy

and s o c o n v e r g e s

also

equivalence

the uniform ,

is defined,

of

b .

is complete.

over

l o c a l uniform

of

G

In p a r t i c u l a r we s e e

more,

Corollary

F

F

to

and s o a l i m i t

the refinement

and l e t

converges

a l s o a Cauchy f i l t e r

X e Xj^ , of

Β

vje u s e e s s e n t i a l l y

e x t e n s i o n theorem t o be f i b r e w i s e

we o n l y h a v e u n i q u e n e s s

of

limits

t h e same

(4.21)

above.

separated,

not

for convergent

filter

106

ι.Μ. James

in a restricted q^H

sense.

converges

converge that

thus

defined

be any e n t o u r a g e

D

that

of

assert

such

Since

on D

A is

Μ X Ν C D

open

and s o

ilarly

φΝ c Γ [ ψ ( η ) J

we h a v e

that

all

D

into of

X)

the

cannot

Yj^ .

To

on

Then

on

Ε

entourage entourage

(<;)χφ)~^Γ

Let

.

I

This

will

F, 6

be

respectively.

Μ ε F,

Since

for

let

X .

ζ , η ,

members

that

show

X ,

an o p e n

in

e X .

to

such

F · F « F c Ε .

ψ

ζ,η

Y

itself

continuous

we may s u p p o s e

Με G

φ,F

such

converges

φΜ c Γ ( ψ ( ς ) ] ,

some p o i n t s

ζ'

and r M,

that

to simη' ε Ν

pairs

(Φ ίζ') ,Φ (n')) , are

(in

exist

.

fiber

contained

φΜ χ φΝ c F .

by d e f i n i t i o n ,

Η

on

be a symmetric

is

where

there

a filter

Then t h e r e e x i s t s

continuity

converging

the

F

maps

e D ,

is

then

uniformly

and l e t

ψ χ ψ

( ξ , η)

in

D η (ΑχΑ)

the uniform

So l e t filters

points

is

Y

that

that then

establish

ψ(ς),

of

H

b e Β

F ο F ο F c Ε .

X

if

t o some p o i n t

t o two d i s t i n c t

Φ,

such

Thus

contained

(Φ (ζ ' ) ,ψ (ζ) ) ,

in

F,

so

(φ (η'),ψ (η))

that

(ψ ( ξ ) ,ψ (η) ) ε F c F r F c Ε , as required

.

Finally is

unique Just

it

follows

on e a c h as

Corollary

complete

wise

dense

(7.15) Let

uniform

subsets

(4.2l)

so t h a t

was d e d u c e d

(7.16).

wise

uniformly

fibre,

(4.22)

a r g u m e n t s o from

from

X^,

Xj^, to

Aj

ψ

is

from

the e x t e n s i o n unique,

(4.21)

as

ψ

of

φ

asserted.

by a f o r m a l

obtain

spaces

of

equivalent

we

that

X^ over ' over

be f i b r e w i s e Β

and l e t

separated A^^,

respectively. Β

then

Xj^

A^ If

is

fibrebe

A^

fibreis

uniformly

orm spaces over a ase equivalent

to

X^

over

Β .

We a r e now r e a d y t o d e f i n e completion

c o n s t r u c t i o n of

a uniform is i

space over

defined,

complete,

the

X„

subspace

b e Β p^F B,

and

of

Β

F

is

converges

to

space,

b.

is

of

Cauchy f i l t e r X^

fibrewise B.

on

be

of

X

X

separated Consider

(b,F) such

as a uniform

,

where

that

space

over

projection. Λ

X„

X

X

function

pairs

Λ

In f a c t

the

Let

completion

space over

consisting

We r e g a r d

of

as f o l l o w s .

Β χ X

as a uniform

Β χ X

version

with canonical

product

a minimal

under the f i r s t

4,

The s e p a r a t e d

The u n i f o r m

and f i b r e w i s e

a fibrewise

Section

Β .

as a uniform

: X ·> X .

107

is

just

the pull-back

^

j*X

of

X ,

regarded

a as a uniform space over j

: Β »• Β

is

Cauchy on

Β ,

refinement

of

canonical.

by d e f i n i t i o n , the

limit

Β

since Nj^ , is

j (b)

For i f

F

is

since

the of

with projection (b,F)

Cauchy on p*F

l i m i t of

ρ

e Xg X .

converges

where

then Also

to

p^F p^F

b .

is is a

Now

which c o i n c i d e s Λ In o t h e r words p ( F ) = j (b)

j*nj^.

j*P*F

,

,

pf , with ,

as

/y required.

This i d e n t i f i c a t i o n

fibrewise

s e p a r a t e d and f i b r e w i s e

X

is

makes

it

clear

complete,

s e p a r a t e d and c o m p l e t e . Λ We r e f e r t o X^ a s t h e f i b r e w i s e

that

over

separated

X„ Β

Β ,

is since

fibrewise

A

c o m p l e t i o n of

X .

Of c o u r s e

Xg c o m e s e q u i p p e d w i t h a

A

canonical function i : X where χ e X, . In f a c t b

X^ o v e r Β , g i v e n by i ( x ) = iX i s fibrewise dense in X , Β

when t h e p r o j e c t i o n o f

i s uniformly

we

X

open,

and s o by

(7.15)

obtain

Proposition function uniformly

(7.17).

over

B,

Let where

open p r o j e c t i o n ,

φ : X X

is

and

Y

be a

a uniform Y

is

uniformly space over

a fibrewise

continuous Β

with

separated

and

I.M.James fibrewise

complete uniform

space

o n e and o n l y o n e u n i f o r m l y

over

B.

continuous

Then t h e r e

function

ψ : Xj, ->· Y Β

Λ Β

such t h a t

fibrewise

ψ1 = φ ,

completion

The f i r s t corresponding be f a c t o r e d uniformly

of

step is result

through

continuous

fined

to a uniformly

using

(7.15)

dense

in

X

t o show,

in the ordinary

iX .

The s e c o n d s t e p i s

function continuous

Xg ,

and i s u n i q u e ,

is

a uniform

p r o j e c t i o n and

Y

is

of

space over

Y

then

φ

ψ : Xg ·»• Y ated,

a fibrewise Β .

If

therefore,

Β ,

by

Y

over

Β

embedded a s a f i b r e w i s e fibrewise

separated fibrewise

In f a c t

the fibrewise

in the following

sense.

tinuous

over

function

over

Β ,

using

(7.17)

continuous ίΦ = Φβ^

dense

Β ,

the projection we s e e t h a t

function shown X

When

separated X

X

X

being

of

there exists

φ^ : Xg ->• Y^

open

below. i > Y

j

complete

dense

subspace

equivalence

X

is

fibrewise

fibrewise can be

uniformly with

Y

is

functorial

are uniform

uniformly

open.

o n e and o n l y o n e Β

the

X .

be a u n i f o r m l y

and

separ-

complete

construction

over

B,

fibrewise

can b e i d e n t i f i e d

φ : X ->- Y

where

over

with uniformly

c o m p l e t i o n of

Let

is

de-

over

t o a uniform

completion

the

embedding

a fibrewise

subspace

can

again.

Β

i n t o which

φ

thus

separated fibrewise

<7. 1 6 ) ,

any f i b r e w i s e

Β

iX

be a u n i f o r m

is

the

to extend

ψ : Xg -<· Y

(7.15)

φΧ

of

that

over

since

space over

can be e x t e n d e d

over

uniform space

exists, by

case,

Y

function

φ : X -v Y

X

uniform

= iX

The e x t e n s i o n

where

j u s t as in the proof

(4.2?)

let

over

X„ i s t h e f i b r e w i s e s e p a r a t e d ΰ and i : X ->• X„ i s c a n o n i c a l . ti

.

In p a r t i c u l a r Β ,

where

exists

such

that

conspaces By

uniformly

Unifonn spaces over a base Here

i

and

fibrewise

j

are c a n o n i c a l .

separated

Finally

there

to pull-backs. where

B'

fibrewise is

λ*Χ

μ

If

of

function

c o m p o s i t i o n of

completion

λ : Β' ->· Β

i s uniform.

continuous

X

X

continuous ψ :

μ : λ*Χ with

over

(X*X)g,

is

Χ

on

such t h a t

j

is

,

therefore

while

ψψ '

is

i

.

The

: X ->• X„ υ

can

function

and t h e n e x t e n d e d

is

is

t o a uniformly

is in

f i b r e w i s e d e n s e in X„ and (λ*Χ),'^, ij , ij

complete.

So we can u s e

to construct

the canonical

of

i

the

identity

that

a uniform

separated

to

ijj'j

and o b t a i n

ψ

fibrewise

ψ

.

Now

and

a uniformly

function ψ'ψ

on

ψ'

equivalence

fibrewise

the

λ*(1Χ)

.

is

B'

.

λ *X , identity We

are mutually

over

completion

is

for

inverse,

Thus

natural

the

with

pull-backs.

In t h e c o u r s e o f

this

section,

observed,

we h a v e made c o m p a r a t i v e l y

structure

of

possible

ρμ = λ ρ '

function

continuous

and f i b r e w i s e

the pull-back

(7.15)

respect

Β

(X*X)g,

i|j'jX*X

apply

continuous,

function

:

B'

where

respect

X*Xg

separated

continuous

over

with

space over

such t h a t

the canonical

Β'

the

φ.

a uniform

i n t h e same way a s b e f o r e ,

Ψ'

of

as

function

fibrewise

(7.15),

φ^

be u n i f o r m l y

over B' . On t h e o t h e r hand λ*(1Χ) λ * ( Χ „ϋ) , since iX i s f i b r e w i s e dense is

to

comes e q u i p p e d w i t h a uniformly

be p u l l e d back t o a u n i f o r m l y φ : λ*Χ -»• λ*Χ,, D

We r e f e r

t h e q u e s t i o n of n a t u r a l i t y

Let

the pull-back

1 uy

the base space.

to develop a similar

a s t h e r e a d e r may h a v e little

This s u g g e s t s theory for

u s e of that

uniform

the

uniform

i t m i g h t be spaces

over

1 ιυ

I.m.jamex

a topological

base

For a c o m p l e t e l y

space,

satisfactory

to proceed to a further are

(in a c e r t a i n

vertically. studied

sense)

These are

stage

has b e e n done i n

theory,

however,

it

is

the fibrewise III

of

horizontally

mentioned.

necessary

uniform

spaces

[11 1.

The r e s u l t s

The work o f Hunt [ 6 ] on u n i f o r m

which

and u n i f o r m

s e c t i o n may s e r v e a s an i n t r o d u c t i o n

general r e s u l t s .

L 8 J.

and c o n s i d e r h y b r i d o b j e c t s

topological

i n [ 9 ] and i n C h a p t e r

in the p r e s e n t

i l s o be

and t h i s

I have given

to these spreads

more

should

8.

Uniform covering spaces

I a s s u m e t h e r e a d e r t o be f a m i l i a r t h e o r y of c o v e r i n g Godbillon [5]

or

spaces in the t o p o l o g i c a l

fully that

of

covering

group, which

theory

there

of

o f c o v e r i n g map.

In t h e uniform

we n e e d t o s t r e n g t h e n lence in a similar

stronger condition

for 1.

R

[22].

This

section

suggests

equivalence

I give

and

l o c a l uniform

that find

equiva-

and c o m p a t i b l e . still.

r e l a t i o n or

2 we h a v e d e f i n e d

the

the

terms

What we now n e e d i s a

Let us say t h a t t o the uniform

R

is

structure,

strongly if

there

the

case,

that

R » D = D » R

each symmetric b a s i c entourage

D .

The a p p r o a c h a d o p t e d by L u b k i n [ 1 5 ]

This i s is

an

difference

t h e o r y we c a n e x p e c t t c of

of

.

an e s s e n t i a l

b e an e q u i v a l e n c e

In S e c t i o n

with respect

a base such

is

[16]

d e a l t w i t h more

the f i n a l

topological

the d e f i n i t i o n

let

X .

weakly compatible

(8.1)

is

fashion.

To s t a r t w i t h , uniform space

local

in

i n Massey

t h e form w h i c h s u c h a t h e o r y m i g h t t a k e ^

between the notion

exists

as

t o d e v e l o p a uniform v e r s i o n

s p a c e and i n

In the t o p o l o g i c a l

compatible,

sense,

t h e c a s e where t h e

[ 3 ] but see a l s o Taylor

ought t o be p o s s i b l e

the notion of outline

the theory concerns

a topological

in Chevalley it

classical

( w i t h some t e c h n i c a l d i f f e r e n c e s )

An i m p o r t a n t b r a n c h o f base space i s

with the

unrelated.

112

ι.Μ. James

for

example,

action

of

if

Β

over

alence

Β

discrete

Proposition

is

just

Β

Now

and l e t

D ·

D

pD[x]

is

X £ X .

In f a c t

entourage

of

injective

(ξ,η)

Β

if

D[x]

open.

s

:

ECb]

a uniform

X

(6.6)

strong

D[x]

onto ρ

to the p a r t i a l

(X

e χ

)

each

on

equithe

and our o b s e r v a t i o n

that

t o show t h a t is

a covering

of

X

p{x), Ε =

where ρ .

E[p(x)],

maps

for

uniformly

be a uniform

of

to

map.

X/G.

<= p D [ x ] .

transverse

a

we h a v e

Dtx] section

a space

covering

satisfying

to the projection

= ρ(η),

as

space of

sense

that

uniform

compatibility.

X

E[p(x)]

Β

covering

c a s e we n e e d

ensures

Therefore

We r e f e r

acting

structure.

covering

space of

(6.6),

b e an e n t o u r a g e

ρ(ζ)

which i s

a s a map o f

E[p(x)].

is

X

discontinuously

a neighbourhood

tinuous, is

from

So l e t

and t h e n

a local

a uniform

transverse

(8.1)

since

£ D · D

D

is

is

equiv-

both

the uniform

ρ

as

the

is

and ρ

the above

sense.

ρ

we d e s c r i b e

implies

in

that

satisfied

properly

the general

topological of

that

a combination of

space

by

to

be a group

X

be a uniform

We a s s u m e

Also,

G

Then

to

covering

such t h a t

is

Let

X .

Returning

space

Β

Τ .

equicontinuity

uniform

are

Χ

determined

course,

and u n i f o r m l y

space

let

a uniform c o v e r i n g

space

(8.2).

continuously

This

of

is

ρ .

^ΔΒ

space of

Β χ Τ

uniform

s p a c e and

and t r a n s v e r s e

When t h e s e

For e x a m p l e ,

in the

(ρχρ)

imply,

equicontinuous

X .

projection

R =

covering

uniform

on

compatible with

equivalence.

uniform

G

with

These c o n d i t i o n s

uniform

d e t e r m i n e d by a u n i f o r m l y

be a uniform

relation

strongly

is

a group

So l e t space

R

ρ

for

(pxp)D

.

each is

e D[x],

Now and i s

ρ

point

an

Moreover ζ,η

(8.1)

is open,

topologically

P1D[X] then consince onto

υ mjorm covering spaces

llj

d e t e r m i n e d by t h e i n v e r s e homeomorphism a s t h e s l i c e through t h e p o i n t cover

χ .

p~^ECb],

some

X ε Xj^

(ς,χ')

,

(χ,χ')

ρ(ξ)

(8.1),

since

ε Xj^ ,

e D ο D

« ECb]

such t h a t

f o r some

as required.

are d i s j o i n t , χ,χ'

if

η ε X

by

ζ = s^,p(5)

and

since

and

ε D ,

Note t h a t t h e s l i c e s ,

if

and s o

(η,χ)

ρ (ξ ) = ρ ( η )

χ ε Xj^ ,

(ξ,χ)

ε D

where

χ = χ'

,

by

each p o i n t s

to X

as above,

b ε Β ,

s'

all

E'[b]

β = ρ(ς)

ε D ,

so

Β

is

uniformly

.

E' c ε

i s connected

the p a r t i a l

χ e X,

X

that

slices

t a k e an e n t o u r a g e

such t h a t

for

for

hence

where

(ξ,χ')

and we can r e s t r i c t

: E [ b ] -<• X ,

e Xj^,

transversality.

We may t h e r e f o r e

E = (pxp)D

ε D

,

From now on we s u p p o s e t h a t t h e b a s e s p a c e locally connected.

χ

and s o

= s^,(B), and

X

for a l l

Also note that d i s t i n c t

ξ = s^(B)

then

then

of

for

sections

We do n o t

assume

D

E' = ( p x p ) D '

f o r some e n t o u r a g e

D'

of

X

such

that

D' c D . Proposition

(8.3).

Let

Y

be a u n i f o r m

a uniform q u o t i e n t s p a c e of form s p a c e s . so i s

If

is

where

X,Y

and

Β

Β

a uniform c o v e r i n g space of

and

are Β

unithen

Y To

see this

ρ = qφ : X ->• Β of

X

X ,

space over

X

let

φ:Χ-«·Υ,

be t h e p r o j e c t i o n s .

which s a t i s f i e s

(φχφ)Ο

is

(8.1)

an e n t o u r a g e o f

respect to

S = (qxq)~^A,

For e x a m p l e l e t equicontinuously the uniform

space

see that

X/H

subgroup

Η

is of

q:Y->B

G

with respect Y

D

which s a t i s f i e s

R = (pxp) (8.1)

then

with

uniformly

properly discontinuously

By c o m b i n i n g

^Δ

transversality.

be a d i s c r e t e g r o u p a c t i n g

on

t h e l a s t two r e s u l t s we

a u n i f o r m c o v e r i n g s p a c e of G .

i s an e n t o u r a g e

to

and s i m i l a r l y w i t h

and u n i f o r m l y X .

If

and

X/G

for

each

I.M.James For t h e r e m a i n d e r

of

this

s e c t i o n we work a l m o s t

i n t h e c a t e g o r y of u n i f o r m s p a c e s o v e r a uniform If

X

is

a uniform

a uniform

covering

space over s p a c e of

unifotm

self-equivalences

uniform

covering

which

t h e y form

(8.2).

(8.4).

and

g^

agree at

φ = g^

connected.

Let

X

Β .

D

transformation

properly

X

i s uniformly

as

required.

ρ .

Let

such t h a t

X ,

: X -«• X

Let

i s uniformly

X

£ D

for

since

are uniformly

be a u n i f o r m

locally

by

such

connected.

that

moreover X ,

G ,

by

(2.7) ,

in t h i s

connected

case,

uniform

the uniform

Poincar^

transverse

be a uniform

c o n n e c t e d we o b t a i n from

(8.5).

,

g £ G

which i s

X

by t r a n s v e r s a l i t y ,

φ , id

X/G

discontinuous.

φ : X

(φ (χ) , χ )

dis-

.

be a u n i f o r m l y

properly

the

transformation,

throughout

Therefore

b e an e n t o u r a g e o f

= X ,

the functions

X

χ

Then t h e a c t i o n of

i s uniformly

to the p r o j e c t i o n

Β

and u n i f o r m l y

the uniform Poincare group

For l e t

where

Suppose t h a t

and t h e n c h o o s e an e l e m e n t

Consequently

s p a c e of

Proposition

X .

covering

precisely

φ(χ)

space

X .

self-equivalences

a uniform

uniformly

Then

be a g r o u p o f u n i f o r m

the

group

P o i n c a r e g r o u p of

is

φ

is

called

and t h e d i s c r e t e

X

is

group of

X

φ : X

Then

X

These are

c o v e r i n g s p a c e of

since

covering

of

Β .

a uniform

·

Proposition

over

equicontinuous

πφ = TT ^ i r g ^

is

X ,

connected uniform

χ e X

B.

we s h a l l n e e d t o c o n s i d e r

is

.

X

if

X

If

- x.g

and p a r t i c u l a r l y

t h e uniform

G

base space

so that

choose a point φ(x)

let

uniformly

continuous,

of

i s called

the uniformly

action is

Β ,

transformation s

For e x a m p l e , of

Β ,

entirely

covering

some p o i n t ρφ = ρ . continuous

(2.7)

that

χ £ X . Since and

since

φ = id

c o v e r i n g space of Then t h e a c t i o n

,

Β , of

unijorrn

the uniform

and

is

E'

c Ε

is

all

transverse

uniform c o v e r i n g

b = p(x)

at

(pxp)D

b ε Β ,

to ,

ρ .

where

D

I assert

then

equicontinuous.

be as b e f o r e

φ : X ->• X .

.

sections

where

satisfies

that

(φ(ς),φ(χ))

transformation

The p a r t i a l

the point

connected,

b ,

and s o

and

if

ε D

(8,1)

(ξ,χ) for

ε D',

each

For c o n s i d e r

(3.12)

of

and s o

ξ ε D[x] = s^E[b]

ρ(ς)

ε E'tbJ,

φ(ς)

ε φs'^E'[b]

(φ(ς),φ(χ)) In t h i s

topology.

thus

ε D

also

=

by t h e

(ρ(ξ),ρ(χ)) .

Now

the

E'[b]

by

ε Ε'

in the topological

and

and

so

φ ,

not

t h e uniform

if

Φ

sense

then Thus i t

is

we h a v e o n l y

continuity.

a covering

Φ is

In

base

covering

a covering

i s unnecessary

spaces

over

used fact

transformation transformation to

distinguish

t h e u n i f o r m P o i n c a r e ' g r o u p and t h e t o p o l o g i c a l

connected

so

Therefore

i t may b e o b s e r v e d ,

sense.

uniform

hypothesis,

as r e q u i r e d .

of

group for

is

coincidence

ζ ε D'[x],

C 0[φ(χ,],

t h e argument shows t h a t

i n the uniform

We h a v e

ς ε s^E'Cb]

argument,

the continuity

ρφΞ' = p s ' , . X Φ IX;

φs^E·Γb] = s^^^jE'Cb],

theorem

between

i s uniformly

sections

where

agree

X

Ε =

D' = D η ( p x p ) ~ ^ E '

partial

j ^

and

c o n n e c t e d for

D « D

where

spaces

Poincare' group on

For l e t E'Lb]

tovenng

a uniformly

Poincare

locally

space.

By c o m b i n i n g

Corollary

(8.6).

covering

s p a c e of

the

last

Let Β ,

X

two r e s u l t s w i t h

be a u n i f o r m l y

where

Β

(8.2)

we

connected

i s uniformly

obtain

uniform

locally

I.M. James connected. form

Then t h e n a t u r a l

covering,

where

G

Among t h e u n i f o r m special

role

following

the

uniform

c o v e r i n g s of

p l a y e d by t h o s e

ττ : X

X/G

is

a uni-

Poincare group of

a given base space

which are regular

in

X .

a

the

sense.

Definition regular

is

is

projection

(8.7).

if,

The u n i f o r m

first,

t h e uniform

X

covering

is uniformly

P o i n o a r ^ group of

X

space

connected

acts

X

of

and,

Β

is

secondly,

transitively

on

the

fibres. For e x a m p l e , of

the uniformly

neutral

of

X/G

Proposition Β .

over

Β .

X e ρ ^ (b)

X

is

Let

this,

choose

ε ρ ^(b)

is

where

be a u n i f o r m l y a uniform a point

exists

b e Β is

φ

Therefore

φ = ψ

Χ ,

Let X

G

be a

through uniform

throughout

at

(discrete)

equivalences.

the diagonal

X X Τ .

If

t h e a c t i o n on

uniformly

properly discontinuous (8.2),the

I T : X X T - > X X q T

action

is

uniform

by

.

X

natural

is

space

function

point

(2.7),

Since

transformation

Then

ρφ = ρ = ρψ .

since

Χ

is

(8.8). on t h e u n i f o r m

Consider,

action

covering

transformation.

and a

group a c t i n g

Τ ,

by

then the

covering χ

This proves

G-space

and h e n c e ,

the

the projection.

a uniform

which agrees with

connected.

If

continuous

covering

ψ : X -»• X

uniformly

X .

be a r e g u l a r u n i f o r m

ρ : X -<· Β

there

space

self-equivalences

regular. X

φ : X ->· X φ

,

be a g r o u p of u n i f o r m

properly discontinuous

Then

To s e e

φ(χ)

by

(6.8).

Let

G

connected uniform

and u n i f o r m l y

covering

of

let

of

G

any

discrete

on t h e u n i f o r m

uniformly then

for

so i s

projection

space

product

equicontinuous

and

the a c t i o n

X χ Τ

on

mjorm covering spaces is

a uniform

X Xq Τ ,

covering.

in the usual

Proposition space

of

let

G

H e r e we h a v e w r i t t e n

(8.9).

Β ,

way. Let

where

(X''T)/G . a s

X

Β

be t h e u n i f o r m

be a r e g u l a r u n i f o r m

i s uniformly

locally

P o i n c a r e group of

covering

connected,

X .

Then

and

the

projection ρ : X is

Τ Η- X/G = Β

a uniform Here

below,

covering,

ρ

is defined

where

natural

for

ρ

is

i·'

the f i r s t

>

projection

and

π

denotes

shown the

space.

X

π

π Χχ^Τ

> X/G

G

ρ

ρ

is

a uniform

are the p r o j e c t i o n s is

Τ .

through the commutative diagriim

p r o j e c t i o n onto the o r b i t

XxT

Now

each d i s c r e t e G-space

a l s o a uniform In f a c t ,

t u t e s a functor

covering, π , by

since

(8.2).

By

is discrete, (8.3)

and s o

therefore,

P

covering.

u n d e r t h e c o n d i t i o n s of from

(8.9),

X x_ G

consti-

t h e c a t e g o r y of d i s c r e t e G - s p a c e s and G-

maps t o t h e c a t e g o r y o f u n i f o r m uniformly

Τ

continuous functions

r e s u l t s w h i c h may h e l p t o

covering over

Β .

illustrate

spaces over

Β

and

We p r o v e t w o f u r t h e r

the behaviour of

this

functor . Proposition s p a c e of Τ

(8.10).

Β ,

where

Let Β

be a d i s c r e t e G - s p a c e ,

g r o u p of onlv

if

X . G

Then

X

be a r e g u l a r u n i f o r m

i s uniformly where

X x^ Τ

acts transitively

G

is

locally

on

Τ .

connected.

the uniform

i s uniformly

covering

connected

Let

Poincar^ if

and

110

I.M. James For

T/G

suppose

that

is uniformly

uniformly given

X x^ Τ

connected,

continuous

there

exists a

orbit.

Thus

Since

X

: Χ x^ Τ

if

t,t'

T/G

t Τ

is a

are

η ,

,

where

t ^^ , t 2 , . . . , t ^

In o t h e r words t h e a c t i o n

Conversely

t

prj/G

Then

chain = f

seme i n t e g e r

since

surjection.

^ = for

i s uniformly connected.

suppose t h a t

i s uniformly

G

is

transitivity

connected,

SO

Τ

implies

in the

so i s

that

same

transitive.

acts transitively

t Τ and h e n c e s o 13 t h e p r o j e c t i o n

However,

lie

Χ x [t},

ΤΓ{Χ χ { t } ) π(X χ { t } )

on

Τ .

for

each

point

C χ x^ Τ .

= X χ _ Τ , and LI

X

i s uniformly

Proposition s p a c e of X/H for

is

(8.11J.

Β ,

every

where

Β

to

subgroup

To s e e k +

X

Let

where

equivalent

this, X

k(x)

continuous

X

connected. X

i s uniformly

of

m -

= (x,e)

,

locally

the

covering Then

space over

Poincare group

the uniformly

proof.

connected.

as a u n i f o r m

t h e uniform

consider G

be a r e g u l a r u n i f o r m

X x^, G/H

Η

This completes

continuous

G

Β ,

of

X .

functions

X , and

m(x,g)

= x.g

.

These induce

uniformly

functions

X/H -<• X x^ G/H ^ X/H G which are m u t u a l l y

inverse.

We a r e now i n a p o s i t i o n t o g i v e q u e s t i o n which o r i g i n a l l y first

stage

universal

is

t o show t h a t ,

covering

space of

in a natural fashion. versions

of

prompted under

an a n s w e r t o

this

conditions,

space i s

The c o n d i t i o n s ,

the corresponding

investigation.

suitable

a uniform

conditions

the

the

uniformizable

which are in the

The

just

uniform

topological

mjorm coverg spaces case,

are as

Definition

follows.

(8.17.) .

The u n i f o r m

pathwise-connected structure

If

if

there

such t h a t for

a path in

Β χ Β Β

connected

is

Β

is

connected

(b,b')

b = bj^ j b j , . . .

s o on. to

b'

which shows

Definition locally

Ε

in the

(3.13).

such t h a t

of

Ε ,

there

p o i n t of

Β

b2

t o b^

Ε .

locally let

is

Ε

pathwise be a

uniformly

f o r some i n t e g e r Β

exists

η .

Thus

we c a n j o i n

by a p a t h i n

Β

is

if

if

b^

Β ,

t h e s e p a t h s we o b t a i n a p a t h from

that

for

uniform

For

If

locally

the

and u n i f o r m l y

e Β .

The u n i f o r m

between p o i n t s

to every

an E - c h a i n i n

then

simply-connected

structure in

b,b'

By j u x t a p o s i n g ,

connected

is

Β ,

uniformly

e x i s t s a base for

e E" ^

= b'

by a p a t h i n

is

pathwise-connected.

and l e t

then

Β

each basic entourage

uniformly

then

space

from t h e d i a g o n a l

b a s i c entourage

b^

l

to and b

pathwise-connected.

space

there e x i s t s

Β

is

a base

each basic entourage the diagonal

is

uniformly for

Ε ,

serai-

the

uniform

every

path

equivalent

t o a path

diagonal.

Entourage

satisfying and ( 8 . 1 3 )

(8.12)

Here and e l s e w h e r e with respect

in this

Entourage satisfyini,· but not (8.13) s e c t i o n we c l a s s i f y

t o homotopy w i t h e n d - p o i n t s

Both t h e above c o n d i t i o n s

kept

are s a t i s f i e d ,

(8.12)

paths

fixed. for

example,

when

1 Β

I.. James is

a compact a b s o l u t e

diagonal

is

is

a uniformly

uniform

the t o p o l o g i c a l

the

converses

of

Returning

Β

b e Β

bg € Β .

We g i v e

Β

λ

uniform

Β X Β

covering

(bj,b2) in

€ Ε

Β X Β

a path in subsets

from

(bg , b g )

a class

Ε .

As

projection

of

ρ : Β continuous

t h e o r y an a c t i o n o f

point from

α £ π^^(Β) (b,a+A)

bg

to

Β ,

ε Β , b .

equivalences.

by a p a t h i n

Ε

Β

,

of p a i r s Β

let

,

a uniform

g i v e n by

ir^^ (B)

bg on

transforms where

E*

b^

to

be t h e

b.

= b ,

path-class

of

Β

on

Β .

is

both

of

Β ,

is defined,

under which

t h e base for

(b,X) λ

ε Β

is a

into

the uniform (λ^,λ2)

properly discontinuous,

·

path-class self-

structure

c a n be

if

is

represented

Moreover since

the the

i s through uniform

(α+λ^^,α+λ2)

the

consist-

topological

and

by

The

As i n t h e

the point

if

that

(8.12).

π^ (Β)

.

subset

such

as a

structure

o p e n by

b ε Β

t h e n so can

where

can be r e p r e s e n t e d

p(b,X)

Β

since

to

(b,X),

fran

regarded

(bj^,b2)

based a t

satisfied.

with respect

( (b^^ ,λ ^ ) , (b2 ,λ 2) )

C l e a r l y the a c t i o n

i s uniformly

are

runs through the e n t o u r a g e s

i n v a r i a n t under t h e a c t i o n

action

of

and u n i f o r m l y

In f a c t

pathwise-

(8.13)

Β ,

paths in

the fundamental group

loop-classes

element

Ε

group

follows.

Ε

pairs

form a b a s e f o r

Consider ing

of

as

to

and

consists

and s u c h t h a t

E*

uniformly

of

semi-locally

simply-connected,

be a

s e t of

Β

structure

consisting

uniform

valid.

Β

(8.12)

the

Β χ Β .

and a u n i f o r m l y

case l e t

For e a c h b a s i c e n t o u r a g e of

then

In t h e c a s e of a t o p o l o g i c a l

Thus is

of

semi-locally

space such that

and w h e r e

retract

two s t a t e m e n t s are

be t h e u n i v e r s a l

a basepoint

is

to the general

connected uniform Let

space

sense.

these

since

locally pathwise-connected

locally pathwise-connected,

simply-connected in

retract,

a neighbourhood deformation

Clearly space

neighbourhood

the Ε

is

an

Uniform covering spaces e n t o u r a g e of

Β

such that

and s i m p l y - c o n n e c t e d

in

{(b,atx),(b,X))

then

path

(v,v)

e E*

in

ΔΒ ,

required.

Β ,

is

for

hence

both

open,

equivalence. by

(8.2),

as

asserted.

so i s

Now

Β

and s o i s

space of

standard

theory

space over

Β ,

where

t o the t o p o l o g i c a l uniform

covering

uniform

structure

is

covering

X

ρ

equivalent to G

s p a c e of

a

B/G

this

every covering

space

space i n t h e uniform

i s a uniform

Β

itselE ,

sense. ,

X .

By t h e

as a

topological

π^(Β)

,

Since

shows t h a t

X

is a

c a n be g i v e n

these

in the t o p o l o g i c a l

isomorphic

B/G

covering

t h a t under

sense.

Β/ιτ^^(Β)

pathwise-connected

s o a s t o become a u n i f o r m therefore,

continuous

s p a c e of

i s a subgroup of

Β

ρ

s p a c e of

as

induces a

i s unifonnly

is

by a

οι = 0 ,

: Β ->- Β

in the t o p o l o g i c a l

P o i n c a r ^ g r o u p of

One may c o n c l u d e ,

conditions

ρ

and t h e r e f o r e

suppose that

Β , X

Since

and s o

a uniform c o v e r i n g

a uniform

More g e n e r a l l y , covering

is

b € Β , and i f

can be r e p r e s e n t e d

ο + λ = λ

ρ ,

pathwise-conriected

each p o i n t

(α+λ,λ)

ρ : Β/π^ (Β) ->• Β .

and u n i f o r m l y

covering

E[bJ

Now t h e p r o j e c t i o n

bijection

Β .

121

space

of

reasonable sense is a

,

Appendix:

filers

In t o p o l o g y nowadays

the v a l u e of

is generally

recognized.

the concept,

i n t h e m a i n p a r t of

an a p p e n d i x

a brief

no d o u b t i t

will

Definition

a c c o u n t of

A filter

F

of

the t e x t ,

the n e c e s s a r y to

non-empty s u b s e t s

theory,

many o f

on a g i v e n

set

of

X

X

a member of

F

(ii)

the i n t e r s e c t i o n

of

subfamily

In a s e t

Such f i l t e r s family for

the most immediately obvious

of

all

s u p e r s e t s of a g i v e n

have t h e p r o p e r t y that

o f members i s

finite

s e t s but i n f i n i t e

s e t form

a l l members o f ment of

a finite

It

is often

family

a filter

F^

F,

F

is

a

non-empty

This

sets contain

is

F^

i s empty.

such t h a t

those

subset. of

always

filters

are

any

the

case

which do n o t

subsets

of

an

the i n t e r s e c t i o n

(A c o f i n i t e

subset

is

of

the

ccmple-

subset). convenient to specify

o b t a i n e d by t a l c i n g

follows .

that

filters

For e x a m p l e t h e c o f i n i t e

of g e n e r a t o r s ,

For t h i s

a non-

the i n t e r s e c t i o n

a l s o a member.

have t h i s p r o p e r t y . infinite

readers. ~

a member o f of

as

F. X

which c o n s i s t

is

of

although

my

such

each s u p e r s e t of

a finite

is

use

I am i n c l u d i n g

(i)

member o f

filter

I h a v e made e s s e n t i a l

a l r e a d y be f a m i l i a r

(Λ.1).

empty f a m i l y

Since

t h e c o n c e p t of

i.e.

all

t o wor)c t h e

a family

a filter

such t h a t

s u p e r s e t s of members of

family

has t o

sati sfy

by d e f i n i n g

the the

filter

a

is

family.

one c o n d i t i o n ,

as

Appendix: filers Definition

(Λ.2).

empty f a m i l y s e c t i o n of

Β

A filter

b a s e on a g i v e n

of n o n - e m p t y s u b s e t s

each f i n i t e

subfamily

Of c o u r s e e v e r y f i l t e r

of

of

B

set

X

X

s non-

such t h a t

contains

c a n be r e g a r d e d

is

tne

a member o f

as i t s

the

filter

of

example,

consider

of

its

a sequence

The e l e m e n t a r y f i l t e r

by t h e f a m i l y

Note t h a t

it

to define

the

the

φ : X

If

defined

by t a k i n g

In c a s e

F

X

is

is

φ a s

In g e n e r a l X

in the

sequence

{Xj^

,...} than the

set

is

X.

the

for

set

filter

k =

ix,^]

we h a v e t o know t h e o r d e r o f

Y

the

on

X.

a subset

φ

Y of

1,2,.. in

ordei

the terms

If,

then a f i l t e r t h e members of

and F

φ

to

however,

in

b a s e on

is

and

φ^F F

Y on

X

are Y

is

as a b a s e .

t h e i n c l u s i o n we r e f e r Y .

G

φ

for

is

G

which the

surjective,

are s a t i s f i e d

φ*0 .

i n c l u s i o n we r e f e r it

X

In c a s e to

φ*0

only defined

X

Y.

non-

then

and t h e is

on

preimages

t h e preimages are a l l

be t h e c a s e when

Of c o u r s e ,

where

hand t h a t we h a v e a f i l t e r

i s d e n o t e d by

the

the filter

a subset

as the trace

cf

of ϋ

when e v e r y member o f

X.

Definition is

of

on t h e o t h e r

as w i l l

meets

X

t h e r e may be members o f

so generated and

on

Images of

c o n d i t i o n s for a f i l t e r

γ

be a f u n c t i o n ,

a filter

are empty.

empty,

F

with the

t o know m o r e

the e x t e n s i o n

Suppose,

G

points

sequence.

sets.

in

of

subsets

necessary

filter;

Now l e t

to

of

For another t y p e

^^^^^

associated

generated

is

supersets.

Β .

own b a s e .

A l s o e a c h n o n - e m p t y s u b s e t c a n be t a k e n a s t h e b a s e o f consisting

inter-

(A.3).

a filter

a member of In t h i s

F'

Let on

F X

be a f i l t e r

on

X.

A refinement

s u c h t h a t e a c h member of

F

is

of also

F'. s i t u a t i o n we s a y t h a t

F'

refines

F,

or t h a t

F

I.. James is

refined

F".

by

When t h e p o s s i b i l i t y

b e e x c l u d e d we d e s c r i b e For

example,

of

let

F

e a c h member o f

and

base

G.

F

Μ η N,

No common r e f i n e m e n t F

and

G

and

filter Y

Y

are

sets.

F

on

X.

such t h a t

Refinement filters

Then

φ*G

φ*φ*Ρ

φ^φ*6

is

is

such

to

Ν e G,

of

that

family

constitutes

which r e f i n e s course,

F

both

if

there

meet.

φ : X ->· Y

be a f u n c t i o n ,

refined

refines

G

by

F,

where

for

each

for each f i l t e r

G

defined.

imposes a p a r t i a l

on a g i v e n

is called

which do n o t

X

Then t h e

and

can e x i s t ,

let

Also

G.

a filter,

is

strict.

be f i l t e r s on

Μ ε F

where

For a n o t h e r e x a m p l e ,

on

G

and s o g e n e r a t e s

a r e members o f

X

and

as

m e e t s e a c h member o f

intersections

a filter

the refinement

F = F'

that

set.

o r d e r on t h e c o l l e c t i o n

A m a x i m a l e l e m e n t of

an u l t r a f i l t e r .

The f o l l o w i n g

the

of

collection

criterion

is

often

useful . Proposition

(A.4).

ultrafilter

the following

if

Μ υ Ν £ F,

For a f i l t e r

where

subsets

N'

Now

is

F'

since

Ν

of

F.

Μ € F'

condition since

F'

of

Therefore

but

F

Μ /

F

then

is

Μ /

Μ u N'

e F

a refinement

of

filter,

since

This ccmpletes

of

F

Μ e F and

n o t an

F

admits a s t r i c t

or

But then

F.

Ν /

is

of

F. The

F'.

strict,

refinement and s o

X - Μ ε

which i s contrary to

Μ ε F'

Ν ε

ultrafilter.

Μ υ (X-M) = X £ F

F,

sufficient:

form a f i l t e r

is

the proof

n e c e s s i t y r e s u l t we o b t a i n

t o be an

and t h e r e f i n e m e n t

X - Μ ε F.

is

X

n e c e s s a r y and

but

F

suppose t h a t

implies that

definition X-M.

such that

on a s e t

then e i t h e r

Μ υ Ν e F

a refinement

Conversely If

X

condition

Μ,Ν c χ ,

For s u p p o s e t h a t

F

and (A.4)

Μ

F'.

the F', the

does not meet

and by i t e r a t i n g

the

Appendix: filers Proposition u n i o n of subsets

(A.5).

subsets is

Let

is

principal.

Since

F^

on

Χ

MsF for

Such u l t r a f l i t e r s

a finite

filter

of

Fq.

Fg

an i n f i n i t e

formed

In f a c t

is

a refinement

or

X - M e F .

some

principal

filter

-If

the

t h e n one of

the

of

F^.

by

set

is

is

X

neither

principal.

necessarily a special

by t h e c o f i n i t e

For i f

X - Μ

(A.5).

generated

X

role

s u b s e t s of

is

any

is

Μ e Fg finite

ultrafilter

then

by

x,

F

either

and s o

This implies that

is

x.

any n o n - p r i n c i p a l u l t r a f i l t e r

But

χ e X - M,

X.

are c a l l e d

set every u l t r a f i l t e r

i s obviously not p r i n c i p a l

refinement

F

g e n e r a t e d by a g i v e n e l e m e n t o f

I n t h e c a s e of

p l a y e d by t h e

a member o f

on

F.

a l w a y s an u l t r a f i l t e r . In t h e c a s e of

be an u l t r a f i l t e r

Mj^, . . .

a member o f

The f i l t e r

F

(x) F

and s o we h a v e a

e F

is

the

contra-

diction . Proposition and is

Y

(A.6).

are s e t s .

an u l t r a f i l t e r For l e t

F

then

of

F

and

then

all

Ν

then

X.

an u l t r a f i l t e r ,

φ^Ρ

is

φφ~^Ν c Ν.

If

φ ^N

i s n o t a menber

Y.

space. forms

Y - Ν G

is

is

is

a

a member o f

a refinement

and s o

therefore

as

The c o l l e c t i o n o f

or

So i f G

a member o f

φ"^(Υ - Ν) = Χ -

Ν

of

φ^Ρ, φ^Ρ

hence c a n n o t be s t r i c t

and s o

asserted.

these preliminaries

in the sense

then

φ'^^Ν

Y - Ν /

χ

X

X

If

The r e f i n e m e n t

of

on

where

Y.

since

of

is a topological

be a f u n c t i o n ,

an u l t r a f i l t e r

Thus, e i t h e r

neighbourhoods of

is

Y

Y.

since

subsets

After X

on

F

Y - Ν £

Ν £ G

is

If

φ : X

be a s u b s e t o f

F.

Ν £ φ*F. φ*F

Ν

Ν « φ^Ρ

member of for

Let

we t u r n t o t h e

For each

situation

χ ε X

a filter

,

the family

Ν

of

the neighbourhood

neighbourhood f i l t e r s

t h a t each neighbourhood

where

of

χ

is

coherent,

contains a

filte:

I.M. James neighbourhood point

of

N'

N'.

on a s e t

X

such that

In f a c t

the

and

filter

is

a neighbourhood

a collection

determines

contains x,

Ν

a topology

(ii)

of

on

filters X if

t h e c o n d i t i o n of

i s then the neighbourhood

of

{

(i)

every

:

χ

e x)

e a c h member

coherence filter

of

of

is

satisfied;

χ

in

the

topology. Definition X.

(A.7).

A point

of

Let

X

is

F

be a f i l t e r

an a d h e r e n c e

point

a d h e r e n c e p o i n t o f e v e r y member o f It fied

is

sufficient,

of

b y t h e members of

point

of

F

then

F

for

a base for

F.

Obviously,

the adherence

points,

just

is

For e x a m p l e , cofinite

topology,

subsets.

F

if

it

is

an

the condition If

χ

is

t o be

an

satis-

adherence

a d m i t a common r e f i n e m e n t . set

of

F,

the i n t e r s e c t i o n

suppose

of

space

F.

course,

and

on t h e t o p o l o g i c a l

that

X

of

is

i.e.

an i n f i n i t e

set

adherence

t h e c l o s e d members o f

in which the c l o s e d

Then t h e a d h e r e n c e

the s e t of

of

sets

set with are the

every f i l t e r

F.

the

finite F

on

X

is

non-empty. Definition X.

(A.8).

A p o i n t of

ment o f

its

F

X

is

F

condition

converges

to

be a f i l t e r

a limit

neighbourhood

When t h i s that

Let

it

is

are i n h e r i t e d

is

if

x.

satisfied

by a p o i n t

Note t h a t

if

of

F.

by r e f i n e m e n t s ,

t h e o t h e r way r o u n d .

is necessarily

F

F

is

a

space

refine-

filter.

t h e n s o d o e s any r e f i n e m e n t points

p o i n t of

on t h e t o p o l o g i c a l

F

χ

converges

In other words, whereas for

To c l a r i f y

we to

the

say χ

limit

adherence

Of c o u r s e a l i m i t p o i n t

an a d h e r e n c e p o i n t .

e X

of

a

points filter

relationship

we p r o v e Proposition

(A.9).

soace

A p o i n t of

X.

Let

F X

be a f i l t e r is

on t h e

topological

an a d h e r e n c e p o i n t o f

F

if

and

Appendix: fillers only i f

it

is

For i f refinement

a limit X

is

of

F

and c o n v e r g e s

to

G

point

of

seme r e f i n e m e n t

an a d h e r e n c e p o i n t o f and t h e n e i g h b o u r h o o d x.

sane r e f i n e m e n t member o f

12"/

G

Conversely, of

F

if

F

χ

is

of

F. '

then the

filter

H^

a limit

F;

common

is

then each neighbourhood

and s o m e e t s e v e r y member o f

adherence point

of

defined

point

of

χ

is a

of

thus

χ

refinement

is

i s an

F.

In t h e c a s e of

an u l t r a f i l t e r

strict

ruled

o u t and s o we d e d u c e Corollary

(A.10).

space

each adherence p o i n t

X

The f o l l o w i n g

For an u l t r a f i l t e r

criterion

of

for

F

F is

on t h e

topological

a l s o a l i m i t p o i n t of

the Hausdorff

condition

is

F

often

useful.

Proposition space

if

(A.11).

and o n l y i f

each convergent {x}

is

the adherence

a Hausdorff

Let

filter

exist

ς,η

η

if

χ

s e t of

X

condition is is

is

a Hausdorff

satisfied

a limit point

of

by

F

then

F.

suppose

be d i s t i n c t ξ

η

converges

that

limit points

there exists

that

are unique

the s t a t e d c o n d i t i o n

p o i n t s of to

ζ

t,

and

suppose that a filter

as a l i m i t point.

every neighbourhood

π

for

with

X.

and s o ,

a s an a d h e r e n c e p o i n t .

n e i g h b o u r h o o d s of Conversely,

ζ,η

F:

in particular,

(A.11),

of admit

following

space

in

space.

To p r o v e

cannot

the

filter

This shows,

fied.

The t o p o l o g i c a l

The by t h e

η

and s o

satis-

neighbourhood condition,

In o t h e r w o r d s ,

there

which do n o t m e e t . seme p a i r of d i s t i n c t

ζ

points

a s an a d h e r e n c e p o i n t £md

Then e v e r y n e i g h b o u r h o o d o f of

is

X

ξ

meets

i s not a Hausdorff

C o n t i n u i t y c a n be n e a t l y c h a r a c t e r i z e d

i n t e r m s of

space. filters.

I.. James as

follows.

Proposition and

Y

(A.12).

are t o p o l o g i c a l

c o n d i t i o n for φ

on

Y

φ : X + Y

spaces.

F

converges

Sufficiency

on

X

to

φ (x) .

i s obvious,

suppose that

converges to .

X

then

But

so φ,Ρ φ(x),

χ

as It

F

continuity at

refines

^φ(χ)'

the

N^,

Ν . .,

at

that filter

F =

and

x.

To p r o v e

χ.

If

hence

o t h e r words

is

image

φ^^F

φ^F

F

refines

by c o n t i n u i t y

Φ ^X ^

X

sufficient

χ e X

χ

i s continuous

where

and

s i n c e we c a n t a k e

refines

χ

refines

F

at a p o i n t

converges to

obtain the usual condition for necessity,

be a f u n c t i o n ,

The n e c e s s a r y

t o be c o n t i n u o u s

whenever a f i l t e r j>,F

Let

at

x,

and

converges

to

required.

is

useful.

in relation to conpactness In f a c t

Definition

conpact

(A.13).

every f i l t e r

on

To r e l a t e

that f i l t e r s

s p a c e s can be d e f i n e d

The t o p o l o g i c a l

X

space

a d m i t s an a d h e r e n c e

this definition

X

are most

as

follows.

i s compact

if

point.

t o t h e more f a m i l i a r

one we

prove Proposition

(A.14).

The t o p o l o g i c a l

and o n l y i f

e v e r y open c o v e r i n g

For s u p p o s e t h a t Γ

of

X

ments of since of

we c o n s i d e r

is

diction,

Γ.

a covering.

of

Γ

covers

X.

that every f i n i t e

section.

Then

base after

taking

Γ*

satisfies

finite

compact

h a s empty

if

subcovering. covering

formed by t h e

comple-

intersection

t h a t some f i n i t e

subfamily

and h e n c e t h e

corresponding

For s u p p o s e , subfamily

is

admits a f i n i t e

Γ*

Then Γ* I assert

X

G i v e n an o p e n

the dual family

a l s o h a s empty i n t e r s e c t i o n ,

subfamily

X

i s compact.

t h e members o f

Γ

Γ*

X

of

space

of

Γ*

to obtain a

has non-empty

the c o n d i t i o n for

intersections,

contra-

a subbase,

and s o g e n e r a t e s a

interi.e. filter

a

Appenix: filers F

on

say

X .

.

By c o m p a c t n e s s ,

Now

particular

χ

belongs

F

a d m i t s an a d h e r e n c e

t o each of

t o t h e members o f

t h e c l o s e d members o f

Γ*.

Thus we h a v e o u r

C o n v e r s e l y suppose t h a t e v e r y open c o v e r i n g a finite

subcovering.

Suppose,

there e x i s t s a f i l t e r the is

intersection empty.

of

F

on

X

the family

Hence t h e d u a l

Γ*

of c l o s u r e s Γ

corresponding

subfamily

Γ*

Proposition F

Let

A

h a s empty i n t e r s e c t i o n . the f i l t e r

F

set

a contradiction,

that

e v e r y member of

the

on

V

of

F

Since

X

where

χ £ X.

X - V X - V

i s compact Now

G

and s o

and from

χ

admits

χ < A

χ e A,

(A.16) .

and o n l y i f

and s o we

of

(A.14).

of

a filter of

Suppose,

A

a filter

G

is

on

G,

of

i.e.

F,

Then X.

χ

the neighbourhood

t h e b a s e of

obtain

F.

an a d h e r e n c e p o i n t

since

to

say, V

of

A

the trace

since

G

w h i c h g i v e s u s our c o n t r a d i c t i o n .

of

refines Hence

obtain

The t o p o l o g i c a l

every u l t r a f i l t e r

When t h e a d h e r e n c e s e t we

A.

i s an a d h e r e n c e p o i n t

( A . 1 0 ) a b o v e we

Corollary

meets

generates

d o e s n o t m e e t t h e members o f

F,

But

Then e v e r y n e i g h b o u r h o o d

of

However

an

Then t h e

be a n e i g h b o u r h o o d

F.

F

F.

For l e t

trace

admits

o f members of

subcovering.

be t h e a d h e r e n c e

X.

X

point.

This completes the proof

on t h e c o m p a c t s p a c e

a manber of

of

of c o m p l e m e n t s forms

a r e a l l members o f

(A.15).

contradiction

Then

a finite

have our c o n t r a d i c t i o n .

in

w i t h no a d h e r e n c e

and s o a d m i t s

Γ*

F ,

that

open c o v e r i n g

t h e members o f

χ ,

to obtain a c o n t r a d i c t i o n ,

family

of

point

in

on

space

X

is

(A.15)

X

i s compact

if

convergent.

consists

of

a single

point

obtain

Corollary Then point.

F

(A.17)

Let

i s convergent

F if

be a f i l t e r F

on t h e c o m p a c t s p a c e

admits p r e c i s e l y

one

adherence

X.

Exercises

1.

Show t h a t

set

Ζ

Ε X X

for

given

of

integers

(η =

1,2,...)

prime

ρ

a uniform

s t r u c t u r e on t h e

i s g e n e r a t e d by t h e s u b s e t s

,

where

t

(ζ,η)

D^

D^

of

i f and o n l y

if

ζ Ξ η mod p*^ . 2.

Let

φ

tinuous for

be a r e a l - v a l u e d

on t h e u n i f o r m

some

e > Ο

continuous, 3.

Let

φ,

continuous

space

and a l l

where ψ

ψ(χ)

function X.

Suppose t h a t

χ e X.

Show t h a t

= (φ(χ))

be r e a l - v a l u e d

on t h e u n i f o r m

which i s u n i f o r m l y

functions

space

X .

4.

t h e uniform space

bounded i f η

for

such t h a t

of

each entourage Η c D"[S]

Show t h a t

the union of

5.

X

Let

and

t h a t the uniform space

Y^

not

discrete.

6.

Let and

D

of

X

spaces,

t o be

there e x i s t s

with

X

an

integer

S

of

Η .

is

bounded.

discrete.

Show

function-

pointwise

an example where t h e uniform

structure

spaces.

is

a uniform

7.

the uniform

structure

ψ

continuous.

said

subset

and

of

Γ:Χ-*·ΧχΥ Show t h a t

is

φ

of uniform c o n v e r g e n c e on t h e

be a u n i f o r m l y

are uniform

X

uniformly

if

i s uniformly

some f i n i t e

be u n i f o r m

and g i v e

uniformly

which are

a p a i r of b o u n d e d s u b s e t s

structure

φ : X ->· Y Y

φ.ψ

c o i n c i d e s w i t h the uniform

convergence,

X

Y

for

is

Sliow t h a t

product

Η

ψ

a e

^ .

are bounded t h e n t h e i r The s u b s e t

|φ(χ)|

con-

continuous

function,

Show t h a t t h e g r a p h

is

where

function

embedding. space

X

is

totally

bounded i f

and

Exercises only

if

all

8.

In t h e uniform

connected uniform

countable

subsets

C^

t o be r e l a t e d

if

η

(ζ,η)

tains

Let

that

for

for

all

11·

D

space

R

Show t h a t

12·

Let

space

X/R

R

Let

uniform relation

X

partitions.

deduce

that

Β

Let

of

that

for

the Euclidean

η

exists

by

C^ .

are

said

an

equivalence

integer classes

connected.

a given point

the u n i o n of is

both

χ

con-

is

the

closed

equivalence

X ,

.

for

every

structure

φ(t)

χ

uniform

,

in

X .

Sj^

(πχπ)

and g i v e

topology.

uniform

a

on

compatible

^S .

defined is

structure .

t X .

relation is

structure

= t3

uniform

on t h e

equivalence

ultrafilter

the uniform

d"IiiJ

on t h e

relation

S„ = κ

the uniform

Show

sets

all

Show t h a t

where

X .

relation

in the quotient

X/R

X

space

o p e n and c l o s e d

equivalence

be g i v e n by

uniform

and

there

X

R[xJ

space

infinite

ξ

the uniform

Show t h a t

φ : ]R -<· »

of

be a c o m p a t i b l e

on

the

connected.

D

class

η ,

be a s e t w i t h

finite

14.

Η

S

and d e n o t e d

of

separated,

and l e t

the quotient

13.

is

called

.

be a c o m p a t i b l e

X ,

equivalence

χ

integers

Suppose

is

uniformly

but not n e c e s s a r i l y uniformly

be a c o m p a t i b l e

X .

,

Show t h a t t h e

be an e n t o u r a g e

positive

χ

the

χ

the p o i n t s

the equivalence

each s u b s e t

Let

X ,

e D" .

t h e component of

10.

of

bounded. ·

all

a given point

each entourage

are c l o s e d

A l s o show t h a t

the union of

and u n i f o r m l y

space

for

are t o t a l l y

canponent

closed

In t h e uniform

thus defined

X

containing

is

such t h a t

subsets

space

(connectedness)

Show t h a t

9.

its

Cauchy

by and

is

not

discrete

Give

the

codomain

t h e domain

]R

the

I.. James uniform is

structure

a strict

i n d u c e d by

refinement

a r e t h e same i n b o t h

15 ,

Dj^ = { ( ζ , η ) i s not

ι b.

structure sequences

cases.

s t r u c t u r e on t h e s e t

X

of

integers

subsets : ζ Ξ η mod η}

(η=1,2,...)

complete.

The m e t r i c s p a c e

and b o u n d e d s u b s e t o f

17.

Show t h a t t h e l a t t e r

o f t h e former b u t t h a t t h e Cauchy

Show t h a t t h e u n i f o r m

g e n e r a t e d by t h e

φ .

X X

Show t h a t t h e r e a l

has the property i s compact.

line

s t r u c t u r e g e n e r a t e d by t h e AIR u ( ( a , " )

X (a,"") )

for

all real

α .

18.

Let

Η

and

X ,

with

Η

c o m p l e t e and

bounded. that

D[HJ

1 9.

Let

JR

Show t h a t

X

D[K]

is

complete.

i n t h e uniform

,

be d i s j o i n t

are

closed

subsets

Κ

completion.

s u b s e t s of

t h e uniform

b o t h c l o s e d and

Show t h a t t h e r e e x i s t s an e n t o u r a g e and

X

i s not complete

Determine t h e

Κ

that every

space

totally D

of

X

such

t h e uniform

space

Show t h a t f o r

each

disjoint.

be t h e s e p a r a t e d c o m p l e t i o n o f

Λ X ,

with canonical

subspace

A

of

X

function

i

: X -i- X .

the s e p a r a t e d c o m p l e t i o n of

equivalent

t o t h e c l o s u r e of

20.

Let

β

that,

in the uniform t o p o l o g y ,

iA

in

is

uniformly

G .

Suppose

X .

b e a u n i f o r m s t r u c t u r e on t h e g r o u p all

A

left

translations

are

contin-

Exercises

13 i

uous wiiile t h e f a m i l y Show t h a t of r i g h t

G

is

uniform

Let

iJ

Show t h a t

G

topology, with

ii .

22.

Let

with

Η

2 3.

is

Η

Let

t h e f^uπily

then

Si

coincid'

topological

G χ G

a topological

Κ

(i)

open or

Η

then so i s

Κ c Η

s i o n of

Η

(ii)

of

Η

g r o u p of

in

K\G

Η the

acceptable

the

such

that

continuous.

t o the

uniform

structures

coincide

t o p o l o g i c a l group

compact.

Show t h a t

G ,

Η π Κ

is

G

K.

Show t h a t

group

G.

Show

Show t h a t

if

Η

is

the t o p o l o g i c a l group

subgroup of

i n d u c e s a uniform

q u o t i e n t uniform is

the t o p o l o g i c a l

if

neutral

Η .

be a n e u t r a l

Η

Let

uniformly

i s a l s o a subgroup.

be a s u b g r o u p of

with the r i g h t in

is

G

group, with respect

be s u b g r o u p s o f

i s an SIN g r o u p t h e n s o i s

let

G

and r i g h t u n i f o r m

be a subgroup of

the c l o s u r e

G

on t h e g r o u p

G . Η

Let

2 δ.

function

and

either

2 4.

K\H

further,

equicontinuous

structure

and t h a t b o t h l e f t

neutral in

in

equicontiruous.

s t r u c t u r e d e t e r m i n e d by t h e

be a uniform

the m u l t i p l i c a t i o n

Η

i s unifonnly

If,

is

structure.

21 ·

that

right translations

a t o p o l o g i c a l group.

translations

with the r i g h t group

of

G .

Show t h a t the

embedding of

structures.

c o m p a c t show t h a t

Η

pair with

H.K = G

topologically

iscmorphic

Show t h a t i f

Κ

G . and

If

K\H

Κ

,

to the s e m i d i r e c t product

i s an SIN g r o u p t h e

left

inclu-

in

in

quotient

K\G , of

G .

be a n o n a a l

Suppose t h a t Η η Κ = {e}

and

t h e image

i s neutral

b e a n e u t r a l s u b g r o u p and l e t t o p o l o g i c a l group

G

(H,Kj so that Η »< Κ . unifonn

subis G

an is

I.M.James structure

on

G/H

structure. than the

is

a refinement

A l s o show t h a t

latter

then

Κ

if

is

of

the right quotient

t h e former

structure

is

an SIN g r o u p and t h e two

uniform

coarser structures

coincide. 26.

Let

G

that

φ(η)

= η

Provide

G

respect

be t h e g r o u p o f for

all

bijections

a l l but a f i n i t e

with the

t o p o l o g y of

to the d i s c r e t e

topology

φ : IN ->• >1

number o f

pointwise of

integers

n.

convergence,

Μ .

such

with

Show t h a t

G

is

n o t a n SIN g r o u p . 2 7.

Suppose t h a t

hood o f

e

which i s

structure. 28.

Let

{e}

G

Let

with

left

of

e

such t h a t of

e

Let

Show t h a t

G

the l e f t

Η

be a t o t a l l y

uniform in

G

{e}

G

V.H c H.U .

Show t h a t

h a s more t h a n although

in

G .

of for

one

G

and

the t o p o l o g i c a l group G , each

a neighbourhood

V

η gUg ^ gtH

Deduce t h a t

trivial

G,

i s dense

bounded s u b s e t

there e x i s t s

uniform

group endowed w i t h t h e

(e)

structure.

neighbour-

complete.

the i n c l u s i o n

and

admits a

or r i g h t

t o a c o n t i n u o u s endomorphism of

Η

and

right

quotient

same

structure.

31 .

Γ

is

be a n o n - t r i v i a l

Κ

o p e n and n o r m a l .

Let

that

for

group

neighbourhood of

is

e

a

in

G

neighbourhood

.

30. Η

G

are both complete,

29.

U

complete

Show t h a t

topology. extension

the topological

G

closed

Show t h a t i f

uniform

structure,

be a Hausdorff

there e x i s t s

such t h a t

be s u b g r o u p s o f

G

subgroup

the

K/Ol η Κ) then

G/H

g r o u p G,

is

topologically

Η

of

with

is complete,

in

the

is complete,

in

the

abelian t o p o l o g i c a l group.

a complete Hausdorff

Γ .

topological

Show

abelian topological

isomorphic

to

Γ/Η f o r

group some

Exercises 3 2.

Let

Suppose

Κ c Η

that

be s u b g r o u p s o f

t h e image o f

Show t h a t

if

G/K

structure

t h e n so i s

33.

Show t h a t

groups i s

3 4.

also

Let

group

Η

G .

uniform

t o p o l o g i c a l group

the i n j e c t i o n

i s complete

quotient uniform

the

G/H .

H/K + G/K

is

in the r i g h t quotient Show t h a t

structure provided

G . conpact.

urvform

t h e same h o l d s i n t h e

Κ

is

the s e m i d i r e c t p r o d u c t of

neutral in

ccmplete

lefi

G .

topological

complete.

be a t o t a l l y b o u n d e d s u b g r o u p o f Show t h a t

structure,

G/H

is complete,

if

and o n l y i f

H\G

Κ

be s u b g r o u p s o f

the

topological

in the right

is

complete,

quotient

in

the

same

group

G .

structure.

35.

Let

Η

Show t h a t

the

and

a c t i o n of

neutral with respect Show t h a t left, in 35. X

the

instead

on

H\G

to the right

corresponding of r i g h t ,

by r i g h t

q u o t i e n t uniform

assertion

uniform

translation

Is

structure.

holds with respect

s t r u c t u r e provided

Κ

to

is

the

neutral

G . Let

G

be a t o p o l o g i c a l g r o u p a c t i n g on t h e u n i f o r m

through uniform e q u i v a l e n c e s .

s t r u c t u r e of ρ

Κ

the topological

: G

G

is

X (xeX)

transverse

if

Show t h a t t h e l e f t

to

and o n l y i f

the e v a l u a t i o n the s t a b i l i z e r

X

space

uniform

function G

of

χ

is

*

discrete. 37.

Let

uniform

G

space

evaluation

Let

X

X

are uniformly

equicontinuous.

and

group a c t i n g

through uniform

functions

i s uniformly

38.

be a t o p o l o g i c a l

Y

on t h e t o t a l l y

equivalences.

Suppose t h a t

c o n t i n u o u s and t h a t t h e

Show t h a t t h e a c t i o n i s

be u n i f o r m

bounded

action

uniform.

s p a c e s over the uniform

the

soace

I.. James Β .

Suppose t h a t

Also suppose φ Xgid

the p r o j e c t i o n

of

X

that

open,

where

φ : X

Y

i s a function

Show t h a t

φ

i s uniformly

open.

39.

Let

Β

be a uniform

s p a c e and l e t

over

Β ,

with closed projection.

X

are complete.

Show t h a t

40.

Let

B·

over

Β ,

with uniformly

of fore 41 .

X

open.

: X XgX + Y Xg X

i s uniformly

of

i s uniformly

be a uniform

X

over

be a u n i f o r m

Suppose t h a t a l l X

i s fibrewise

s p a c e and l e t

X

open p r o j e c t i o n .

space

the

fibres

ccmplete.

be a u n i f o r m If

a r e ccxnpact show t h a t t h e p r o j e c t i o n

Β .

all

the

i s closed

space

fibres

and

there-

proper. Let

X

be a uniform

p r o j e c t i o n and l e t a l o c a l uniform tinuous function i s uniformly s e c t i o n of

Y

space over

be a u n i f o r m

equivalence. over

open in

Β . X x^Y

Let

Β

with uniformly

space over φ : X

Y

Β

with

open projection

be a u n i f o r m l y

Show t h a t t h e f i b r e w i s e g r a p h o f (the fibrewise

graph i s

the ordinary graph w i t h the f i b r e w i s e

the

uniform

conφ

interproduct).

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Topologies for

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Math.

Theory of

104

Brace 17.

groups.

On n e u t r a l

Une c l a s s e

invariante,

d'espaces C.R.

homogSnes p o s s e d a n t

Acad.

Sci.

(Paris),

238

une (1954),

553-4. 21.

22.

W. R o e l c k e and S. D i e r o l f , Topological

Groups and t h e i r

(New Y o r k ) ,

1981.

R.L.

Taylor,

groups, 23.

Uniform

J.W.

Tukey,

o f Math.

Amer. Math.

Convergence Studies

2,

on

Q u o t i e n t s . McGraw

C o v e r i n g g r o u p s of

Proc.

Structures

non-connected

Soc.

5 (1954),

and U n i f o r m i t y

P r i n c e t o n Univ.

Hill

topological

753-68.

in T o p o l o g y . Press

Ann.

(Princeton)

1940. 24.

J.

de V r i e s ,

Universal

topological

G e n e r a l T o p o l o g y and i t s 25.

A. W e i l ,

Sur l e s

transformation

Applications,

5

(1975),

E s p a c e s δ S t r u c t u r e Uniforme

T o p o l o g i e G d n e r a l e . Hermann

(Paris) ,

groups,

1938.

et

107-22.

sur

la

Index

absolute

neighbourhood

acceptable action

pair

retract

83,

function

133

93

additive

group s t r u c t u r e

additive

uniform

Arens

basic

34-46,

set

126,

viii,

base

22-24,

120,

123,

56,

57,

60,

64,

79,

40,

44,

50,

64,

65,

71,

7-10,

25,

39,

50,

80,

119

129 4,

11,

31,

74,

bounded f u n c t i o n

130

bounded i n t e r v a l

39

bounded m e t r i c

7

bounded s u b s e t

130,

Bourbaki

vii

canonical

function

cartesian

product

cartesian

square

2

8,

31,

79,

83,

89-92,

87,

88,

121,

134

132

65-69,

107-109

20

10,

Cauchy c o n d i t i o n Cauchy f i l t e r

51,

126-129

129

39,

entourage

category

42

89

3-8,

bijection

73

structure

adherence point adherence

120

50, 50-66,

37, 51,

79, 54,

86-87,

92, 60, 97,

98, 78, 104.

117 86, 105,

97 131

73,

74,

80

itu

ijvi.james

Cauchy s e q u e n c e chain

2,

5,

7,

closed ball

50,

54,

27,

103,

41,

function

31,

closed

interval

78

closed

set

closure

132

118,

119

94

closed

closure

78,

56,

136

29,

30,

41,

57,

131-133

29-36,

41,

45,

62,

68,

operator

coarsening

29,

3,

cofinite

30,

coherence

16,

32,

22,

122,

40,

32,

132

41

33,

41,

46,

47,

20,

38 structure

15,

23,

28

compact

56,

94,

95,

120,

42-46,

compatible

24-28,

complete

50-63,

68, 41,

50,

composition

1,

connected

64, 2,

homomorphism

86,

131

104,

105,

132,

135

105,

127,

128

84,

85,

48 97,

30,

51,

107,

132

59,

128

5

covering

space

111,

dense

36,

41-42, 10,

101,

119

60,

12,

91,

57,

64,

87,

134

78,

104,

121

57,

1-6,

37,

79-82,

55,

countable base

7

80,

90,

46

continuous

diameter

81,

132-135

30

function

action

63,

79,

68,

continuous

42,

59,

47,

38-40,

convergent

79,

66-72,

regularity

completion

diagonal

134

125

coinduced uniform

diagonal

62,

126

coincidence

complete

51,

62,

16,

92,

21,

116

66-68, 25,

40,

134 44,

46,

94,

Index direct

14]

product

80,

84

d i s c r e t e pseudometric discrete

topology

discrete

uniform

5 29-33,

38,

structure

73,

75,

92,

95,

116,

3-13,

21,

33,

38,

50,

94, dyadic

rationals

equicontinuous

50, 89,

equivalence

class

equivalence

relation

52,

2,

27, 2,

function

euclidean metric space

euclidean

topology

euclidean

uniform

factor fibre

62,

98,

123

59, 6,

63,

13,

85,

70,

16,

90,

81,

24-28,

85, 41,

90 59,

131

55

95 73 structure

function

group

131

92 54,

euclidean

extension

74,

133

81,

evaluation

73,

116

filter

equivariant

117,

55,

136

48

elementary entourage elementary

112,

117,

4,

8,

37-39,

8 9-94

68,

106-108,

123

81 102,

104

fibrewise

complete

104-109,

fibrewise

completion

fibrewise

dense

fibrewise

property

fibrewise

separated

fibrewise

totally

bounded

102

fibrewise

uniform product

99,

99,

136

107-109 105-109 100, 100,

103 101,

105-109

101

73,

131

63,

67,

70,

142

.Μ. James

fibrewise

uniformly

filter

2,

4,

42-44,

finite

covering

18,

finite

subset

free

6-9,

action

function

56,

71-73,

79,

122-129

27,

54,

68,

75-77,

81,

93,

102

28,

53,

66,

74

117

89

G-space general

53,

103

38 18,

51,

37,

G-set

50,

102,

96

space

functor

connected

94 linear

graph

130,

group

70

group 136

group completion

Hausdorff

87

35,

homeomorphism

identity

73

42,

57,

62,

31,

77,

113

relation

induced topology

1,

70

31,

32

78,

127,

16,

20,

induced uniform

structure

infinite

30,

33,

93,

122,

11-14,

17,

32,

82

set

injection interior

63,

66

invariant

base

invariant

subspace

inversion

71,

Isbell isometry

Kuratowski

90 95

72,

77,

vii 9

29,

30.

34

81,

86

126

134

23,

26,

68,

132

Index left

Cauchy

left

coset

left

inverse

left

quotient

set

left

quotient

space

left

quotient

uniform

left

relation

left

translation

left

uniform

limit

local

77,

87

70,

81

14,

17,

21

70 83 structure

76,

135

81

structure 42,

71-80,

51,

connected

56,

57,

6-9,

30,

55,

m i n i m a l Cauchy

filter

multiplication

71,

multiplicative

90,

60,

133,

61,

134

54,

79,

105,

126,

127

46

uniform e q u i v a l e n c e

metric

133,

70-72

point

locally

'

14

14,

101,

112,

136

132 62-66, 72,

86,

77,

86,

87,

104-107

41,

81,

82,

91,

111,

44,

51,

60,

63,

65,

133

group s t r u c t u r e

74

1

natural projection natural

uniform

neighbourhood

2,

24-27,

structure

filter

I

G-space

neutral

action

34,

42,

neutral

subgroup

normal

space

116

84-86,

133,

135

44

normal subgroup

open b a l l

91 90,

39,

81-84,

117

94

125-128 neutral

116,

133,

73,

94

open c o v e r i n g

42,

94,

128

open f u n c t i o n

31,

37,

38,

134

41,

90

72-75,

87,

. James o p e n homomorphism open i n t e r v a l open s e t

82 55

34-45,

orbit

90

orbit

space

61-64,

91,

106,

129,

131,

133

59,

67,

117

orthogonal

group

orthogonal

transformation

Page

83,

94 9

vii

pathwise-connected

119-121

pointwise

Cauchy

pointwise

complete

pointwise

convergent

Poncet

83

precompact presque

52 59 58

73,

68

invariant

principal

52,

83

filter

50,

63,

product topology

35,

80

proper

function

property

S

136

39,

pseudcmetric pull-back

40

4, 98,

6,

99,

9,

22,

104,

topology

quotient

uniform

structure

rational

line

8,

line

real

n-space

refinement reflexive

4,

3,

41,

38,

8,

33,

8,

73

22,

relation

30-32,

39

107-110

quotient

real

125

42,

85,

90

24-27,

41,

85,

54 37-39,

51-60, 1,

2,

55,

63, 6

73,

64,

132

79,

124-127,

132

91,

98,

133

Index regular

covering space

regular topological relation

1,

relative

topology

relative

uniform

relatives

space

32,

43-47,

94,

95

85,

86,

80

structure

17,

18,

40

42

1

r i g h t Cauchy

78,

79,

right

coset

70,

right

quotient

set

right

quotient

space

right quotient relation

right

translation

r i g h t uniform

86,

70 81 structure

76,

semidirect

product

semigroup

86

separated

6,

91,

135

74-79,

82-85,

7,

22, 107,

33,

83,

separated quotient

139

uniform

57,

62,

65,

67,

68,

77,

86,

87,

97,

104

132

68,

complete

69,

86,

space

96,

97,

27,

67

132

54

39

simply-connected SIN g r o u p

119,

75, 7,

39,

84-86, 40,

121 90-92,

50-56,

60,

134 65,

67,

94 refinement

strongly

84,

135

35,

131,

separated completion

sphere

134-135

83

105,

small

77,

structure

vii,

Sierpinski

81,

70-72

Roelcke

sequentially

87

85

uniform

right

strict

35,

2

1,

reverse

116-118

compatible

3,

124, 111

127,

132

68,

79,

100,

I.. James surjection symmetric

2,

9-14,

relation

82, 1,

topological

embedding

topological

equivalence

topological

group

topological

invariant

topological

isomorphism

topological

Poincarfe

topological

product

topological

space

topological

square

topology

29,

32 31,

39,

79,

133,

134

115

uniform

convergence

7-9,

41,

45,

18-21, 84,

94,

transitive

property

16,

transitive

relation

2, 94,

trivial

topology

trivial

uniform

93,

134

52,

59,

74,

78,

92

27,

40,

53,

54,

56,

60,

91,

102,

63,

68,

132-135

66

118

6 133

112-114 5 32,

structure

tubular neighbourhood

structure

78,

26

95,

29,

two-sided property

74,

96

13-15,

pseudometric

60,

116,

function

trivial

86,

56-58,

action

89,

51,

convergence

transitive

two-sided

83,

40

81,

transverse

80,

30

bounded

translation

55

29

of

translation

37

35

topology

17,

6

group

of pointwise

trace

2,

69-89

topology

totally

123

73

75,

3-8, 120

76

73,

92,

16,

134

32,

50,

55,

73,

94

75-77

inuKx ultrafilter

14" 54,

uniform

G-space

uniform

action

uniform

convergence

50,

58,

92,

97

92,

69,

124-133

96 52

uniform covering space

111-118,

uniform

covering transformation

uniform

embedding

uniform

equivalence

17-19,

56,

8-11,

121

114,

116

67,

14,

74-76,

120,

68,

18,

80,

22,

80-85,

83,

25,

89-91,

108,

37, 94,

130,

62, 98,

67, 99,

133 68,

1C9,

116,

135 uniform

invariant

9,

76,

uniform neighbourhood

100

34-43,

50-53,

uniform

P o l n c a r 6 group

114-118

uniform

product

25,

20-23, 99,

107,

28,

36,

112,

66,

41,

structure

of p o i n t w i s e

uniform

structure

of uniform

convergence

uniform

topology

32-38,

45-52,

uniform

transformation 43,

uniformly

Cachy

52,

uniformly

closed

uniformly

complete

uniformly

connected

convergence

41,

group 44,

58,

80,

51,

75

92,

93,

51,

72,

85,

58, 91,

74,

78

131

89-97

47,

53,

51,

116

uniform

uniformization

72

94,

118

58

59 58,

59

7-12,

18-23,

102,

114-116

52,

27,

uniformly

convergent

uniformly

equicontinuous

uniformly

locally

compact

uniformly

locally

connected

uniformly

locally

pathwise-connected

35-38,

77,

58 89-96,

98,

102-114,

40,

46,

133

46 39,

114-118

119-120

81,

85,

91,

98,

I.. James uniformly

open

10-21, 121,

132,

uniformly

semilocally

universal

covering

set

universal

covering

space

Urysohn

weakly Weil

25,

37,

vii

41,

simply-connected

24-26,

81,

91,

94,

136

120 118,

48

compatible

38,

41

121

119,

120

99-110,

11