Preface
In theoretical and applied areas of mathematics we frequently deal with sets endowed with various structures. However, it may happen that the consideration of a set with a specific structure, say topological, algebraic, order, uniform, convex, et cetera is not sufficient to solve the problem posed and in that case it becomes necessary to introduce an additional structure on the set under consideration. To confirm this idea, it will do to recall the theories of topological groups, linear topological spaces, ordered topological spaces, topological spaces with measure, convex topological structures, and others. This list is not complete without adding the theory of bitopological spaces and also the theory of generalized Boolean algebras connected with certain classes of bitopological spaces. The notion of a bitopological space (X, ~-1,T2), that is, of a set X equipped with two arbitrary topologies T 1 and T2, was first formulated by J. C. Kelly in [151]. Kelly investigated nonsymmetric distance functions, the so-called quasi pseudometrics on X x X, that generate two topologies on X that, in general, are independent of each other. Previously, such nonsymmetric distance functions had been studied in [262] and in [219]. Although [151] is beyond any doubt an original and fundamental work on the theory of bitopological spaces, nevertheless it should be noted that both the notion of a bitopological space and the term itself appeared for the first time in a somewhat narrow sense in [181], [182] as an auxiliary tool used to characterize Baire spaces. For this use, the topologies T1 and ~-2 on a set X, one of which was finer than the other, were connected by certain other relations as well. Mention should be made of the viewpoint of A. A. Ivanov [137], following which a pair (X, T), where X is any set and ~- is any topological structure on X x X, is called a bitopological space in the general sense. Therefore to a bitopological space (X, ~-1,~'2) in the sense of Kelly, there corresponds a bitopological space in the general sense of the type (X, T), where ~- = T1 X 7 2 is the product topology on X x X, which in [137] is called a decomposable bitopological structure (see also [139], [141], and the interesting work [143] together with the bibliography on bitopological spaces [laS], [140], [142]). We shall adhere to Kelly's notion which at present seems to be more flexible for various usage the main objective of this monograph - developing the theory of bitopological spaces with its applications. Distance functions, uniformity, and proximity are the related notions in defining the topology and, naturally, the situation treated in [151] is by no means the
x
Preface
only way leading to a symmetric occurrence of two topologies on the same set; the investigations of quasi uniformity [190], [250] and quasi proximity [203], [124], [243] also lead to an analogous result. These topics are best covered by H. P. A. Kiinzi in [157]. Keeping in mind the symmetric generation of two topologies on a set, along with the above-mentioned cases, we can also consider ordered topological spaces [191], [208], [5], [53], [177], [178], partially ordered sets [10], and hence directed graphs [64], [65], semi-Boolean algebras [212], S-related topologies [252] and so on. On the other hand, there are many examples of nonsymmetric occurrence of two topologies on a set, particularly in general topology, analysis, and potential theory, as well as in topological convex structures (see, for example, [1], [2], [4], [7], [21], [22], [67], [260], [125], [44], [255], [172], [173]). From the above-said it follows that due to the specific properties of the considered structures two topologies are frequently generated on the same set and can be either independent of each other though symmetric by construction or closely interconnected. Certainly, the investigation of a set with two topologies, interconnected by relations of "bitopological" character, makes it possible on some occasions to obtain a combined effect, that is, to get more information than we would aquire if we considered the same set with each topology separately. If we compare all the results available in the theory of bitopological spaces from the general point of view, we shall find that in different cases two topologies on a set are not, generally speaking, interconnected by some common law that takes place for all bitopological spaces. However if, when defining a bitopological notion, the closure and interior operators are successively applied in an arbitrary initial order to the same set, then, in general, these operators will interchange in topologies as well. As a weighty argument in favour of the above reasoning, we can consider the natural bitopological space (R,a~l,aJ2), where R is the real line with the lower wl = {2~,R} U {(a,+oo) : a e R} and upper w2 = {2~,R} U { ( - o c , a) : a e R} topologies [31] playing nearly the same role in the theory of bitopological spaces as R with the natural topology co = { ~ , R } U {(a, b): a, b E R} plays in general topology and analysis as a whole. Indeed, if for an arbitrary subset A c R we take its interior in the topology c01 (respectively, co2), then the smallest closed subset, which contains this interior, is the closure of this interior in the topology co2, but not in wl (respectively, in a;1, but not in w2). Now, if for an arbitrary subset firstly we take its closure in the topology CO1 (respectively, u;2), then the largest open subset, which is contained in this closure, is the interior of this closure in the topology co2, but not in col (respectively, in u;1, but not in w2). This simple example confirms the essence of closure and interior operators, which to each subset A c R put into correspondence respectively the smallest closed set containing A and the largest open set contained in A, on the one hand, and confirms convincingly the above-mentioned interchange principle, on the other hand. In addition to our motives for studying a bitopology, that is, an ordered pair of topologies (T1, T2) on a set X, we have also derived a stimulus from G. C. L. Briimmer [50], where important problems of the same kind are referred to, in particular, hyperspaces and multivalued functions [242], [27]; function spaces [207]; H-closed,
Preface
xi
almost real-compact, nearly compact and k-compact spaces [114], [164], [231], [232]; Wallman compactifications [46], [231]; topological semifields [135]; algebraic geometry and continuous lattices [50]. It should be also noted that at present there are several hundred works dedicated to the investigation of bitopologies; most of them deal with the theory itself, but very few deal with applications. These latter papers have been published after the late 70s (see, for example, [208], [5], [53], [64], [65], [173], [50], [242], [27], [207], [114], [164], [231], [232], [46], [135], [25], [89], [90], [93]-[103], [9]). We should mention [109], [110], where J. Ewert, shows that a separable Banach space with the weak topology and the topology determined by the norm, has interesting bitopological properties, on the one hand, and gives the Baire classification of multivalued functions of topological to bitopological spaces, on the other hand, and [249] since according to its author J. Swart the axiomatic topological characterization of Hilbert spaces is due to a large extent to the bitopological analogue of the notion of an open cover from [113]. In the above context we can also recall [258], where the term "consistent" equivalent to "bitopological Hausdorff" is one of the key notions, and [173], where the "bitopological boundary" is essentially used for establishing the minimum principle for finely hyperharmonic functions. The theory of bitopological spaces and its applications owe much to J. M. Aarts [3], [4]; D. Adnadjevid [5], [6]; S. P. Arya [16]-[19]; B. Banaschewski [23]-[251; T. Birsan [301-[33]; G. C. L. Briimmer [46]-[52]; A. Csgszgr [69], [70]; M. C. Datta [72]-[74]; J. Dei~k [76]-[78]; D. Doitchinov [80], [81]; P. Fletcher [112]-[117]; M. Jelid [144]-[147]; Y. W. Kim [152], [153]; H. P. a. Kiinzi [154]-[159]; E. P. Lane [165][167]; M. Mrgevid [3], [4], [183]-[188]; M. G. Murdeshwar [189]; S. a. Naimpally [189], [194], [195]; C. W. Patty [202]; W. J. Pervin [203]-[205]; H. A. Priestley [208]; I. L. Reilly [213]-[218]; S. Romaguera [221]-[225]; M. J. Saegrove [228]; S. Salbany [229]-[233]; A. R. Singal [235]-[238], [240]; M. K. Singal [235], [237]-[239]; J. Swart [248], [249], and to many other authors, who are not listed here and to whom we offer our apologies. This monograph is a versatile introduction to the theory of bitopological spaces and its applications. It considers the topics of bitopology that were studied perfunctorily or not studied at all and presents original results and examples which, we dare think, will stimulate the reader to further research. The monograph consists of eight chapters, of which Chapters III, IV, V, VI, VII form the core because they contain the basic results related to the abovementioned topics. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, theory of ordered topological spaces, and graph theory. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study an algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces.
xii
Preface
To conclude the preface, we would like to note that we firmly believe that from the standpoint of applications the theory of bitopological spaces has no less promising prospects than the theory of topological spaces. The areas of such applications are, in our opinion, the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, analysis, mathematical logic, the potential theory, the probability theory and many other areas, including those of applied nature. In particular, the study of strong and weak topologies in analysis, the initial and the Alexandrov topologies on a manifold in the global Lorentzian geometry, cohomologies of spaces with two topologies, and the theory of foliations seems very promising for future research.
CHAPTER 0
Preliminaries Besides being auxiliary, this chapter also contains the internal characterization of pairwise completely regular bitopological spaces. In Section 0.1, along with the symbols and notations, we give a survey of the basic concepts and results from the theory of bitopological spaces to be used in our further investigation. In particular, we recall various kinds of pairwise separation axioms and their interrelations established by J. C. Kelly [151]; E. P. Lane [166]; M. G. Murdeshwar and S. A. Naimpally [189]; J. Swart [248]; I. L. Reilly [215], [217]; Y. W. Kim [152]; T. Birsan [31]; A. R. Singal [236]; M. K. Singal and a. R. Singal [238]; C. W. Patty [202]; D. N. Misra and K. K. Dube [180]; M. J. Saegrove [228]; W. J. Pervin and H. Anton [205], and others. Since the study of relations between the theory of bitopological spaces and some other branches of mathematics in Chapter VII demands special knowledge of bitopologies, we recall the appropriate notions of pairwise compactness, pairwise local compactness, pairwise paracompactness, pairwise local LindelSf property, and pairwise paraLindelSf property. These notions were formulated for the first time by P. Fletcher, H. B. Hoyle, and C. W. Patty [113]; J. Swart [248]; M. C. Datta [72]; M. Mrgevid [183], [184]; R. A. Stoltenberg [245]; I. L. Reilly [214]; T. G. Raghavan and I. L. Reilly [209]; T. Birsan [30]. We also present bitopological versions of connectedness and similar type properties, the study of which was initiated by W. J. Pervin [204]; H. Dasgupta and B. K. Lahri [71]; J. Swart [248], and C. Amihg~esei [12]. Consideration is given to the bitopological notions of continuous, open, closed, and homeomorphic maps introduced by J. Swart [248] and A. R. Singal [2361. In the topological case the complete regularity in internal terms, that is, without using the notion of a function, was characterized by O. Frink [119], E. F. Steiner [244], and V. I. Zaicev [264]. Their modifications for bitopological spaces were studied by Saegrove, who used the generalization of Steiner's method, and by us with the aid of the generalized method of Frink and Zaicev. 0.1. S y m b o l s a n d N o t a t i o n s . Basic C o n c e p t s of B i t o p o l o g y Throughout the book, along with the generally accepted symbols, we use our own notations and those from [68] and [200]. Sets are usually denoted with italic capitals A, B , . . . and elements of sets with lower case italic a, b, . . . . Sets whose elements are sets are called families of sets, while their elements are called members. Families of sets and classes of functions,
2
0. P r e l i m i n a r i e s
with rare exceptions, are denoted with one or two script letters A, B , . . . , followed by one italic capital in brackets for families of sets signifying spaces, and two italic capitals in brackets, in the case of classes of functions, denoting the corresponding spaces. The empty set is denoted by ;~, while the symbols N, Z, Q and R are respectively used for the sets of all natural numbers (excluding zero), of all integers, of all rational numbers, and of all real numbers. We also use oc to indicate "an infinite number". Other standard symbols are defined for each n E N as follows: n - 1, k means that n E { 1 , 2 , . . . , k } and n - 1, oc means that n E { 1 , 2 , . . . }. The closed (open) real segment joining a and b, denoted by [a, b] ((a, b)), is the set { x c R " a_<x_
0.1. Symbols and Notations. Basic Concepts of Bitopology
3
Their interest was naturally preconditioned by the essence of many topological and bitopological properties, especially by the axioms of separation. Further, the reasons connected either in an obvious or a veiled manner with the investigation of bitopologies (7"1,7"2) on a set X, where 7"1 and 7"2 are either independent of each other or interconnected by the inclusion, S-, C-, and N-relations or by their various combinations (see Chapter II), on the one hand, and, with applications of bitopologies, on the other hand, led us to the following basic objectives regarding two general cases: (1) to establish pairwise properties using the properties of 7"1 and 7"2 (or the properties of one of them) or other pairwise properties (or their combinations); (2) to establish properties of 7"i using the properties of 7"j or pairwise properties (or their combinations). As our study shows, (1) suggests the development of the theory of BS's, while (2) is natural and typical of applications, especially when i = 2, j = 1, 7"1 C 7"2. Finally, please note that all bitopological generalizations in this work are constructed in the commonly accepted manner so that if the topologies coincide, we obtain the original topological notions. Let (X, 7"1,7"2) be any BS and A c X be its any subset. Then 7"i clA and 7-i int A denote, respectively, closures and interiors of A in the topologies 7"i. If A = {As}sos c 2 x is any family, then i-C1A - {7"i cl As }s~s,
i-Int A - co i-C1 co A - { 7"i int As }ses'
(i, j)-C11nt at - { 7"i cl 7"j int As } scs' (i,j)-IntC1A - co(i,j)-ClIntcoA
- {7"~int 7"j clAs}s~s ,
i-~-~ (X) - { A E 2 X" A is an i-Fo-set } and i - G s ( X ) - coi-)r~(X) - {A 6 2 X" A is an
i-a~-set}.
Let (I~,wl,a;2) and (R, cv) be respectively the natural BS and the natural TS. If I - [0, 1] - {x c R " 0 < x < 1} is the closed unit segment, then (I,c~,a;~) and (I,a;') are, respectively, the BsS of (R, czl,a;2) and the TsS of (R,c~). Moreover, ({0, 1}, cJ') is the two-point TsS of (R, cJ) and (R, ~) is the extended natural TS. D e f i n i t i o n 0.1.1. A subset A of a BS (X, 7-1,7-2) is termed p-open in X if A - A1 [J A2, where Ai E 7-i. The complement of a p-open set in X is p-closed in X, that is, a subset B of a BS (X, 7-1,7-2) is p-closed in X if B - B1 N B2, where B~ ~ co ~ [S4]. Thus a subset A c X is p-open (p-closed) in (X, 7-1,7-2) if and only if A 7-1 int AU7-2 int A (A - 71 clAnT"2 clA). The family of all p-open (p-closed) subsets of a BS (X, 7"1,7"2) is denoted by p - O ( X ) ( p - C I ( X ) ) . It is clear that 7"1 [J 7"2 C p - O ( X ) (co 7-1U co7-2 C p - e l ( X ) ) and so in a BS (X, 7-1 < 7"2), we have p - O ( X ) (p-el(x)
-
The notion of a p-open (p-closed) set is equivalent to the notion of a quasi open (quasi closed) set given by D a t t a [72].
4
0. P r e l i m i n a r i e s
Definition 0.1.1 immediately implies that an arbitrary union (intersection) of p-open (p-closed) sets is p-open (p-closed), while the finite intersection (union) of p-open (p-closed) sets need not be p-open (p-closed) ([72, Remark 7]). Recall that for a subset A of a TS (X, T) the derived set of A, that is, the set of all accumulation points of A, is denoted by A d and A i = A \ A d is the discrete set of all isolated points of A. Hence to avoid confusion with our notation, for a BS (X, T1, T2) we shall use the following double indexation: A d - {x c X " x is an/-accumulation point of A} and Aji _ {x c A" x is a j-isolated point of A} so that the lower indices i and j denote the belonging to the topology and, therefore, always i, j E {1, 2}, while the upper indices d and i are fixed as the accumulation and isolation symbols, respectively. Thus for a BS (X, wl, T2), we have:
Aji = A \ Aj,d A is a j-discrete set ~ A is a j-dense in itself set ~
A - Aj,i
A c Ajd
and A isaj-perfectset
~A-Aj.
d
D e f i n i t i o n 0.1.2. Let (X, T1, "/-2) and (Y, Yl, Y2) be BS's. Then a function f : (X, T1, ~-2) --4 (Y, ~/1, ~f2) is said to be/-continuous (/-open,/-closed, an i-homeomorphism) if the induced functions f : (X, 7i) ~ (Y, 7i) are continuous (open, closed, homeomorphisms)[204], [236]. The classes of all/-continuous,/-open,/-closed functions of X to Y are denoted by i-C(X, Y), i-O(X, Y), i-Cl(X, Y), respectively, and the classes of all i-homeomorphisms of X to Y are denoted by i-7-l(X, Y). It is obvious that
d-C(X, Y) - 1-C(X, Y) A 2-C(X, Y), d-O(X, Y) = 1-O(X, Y) N 2-O(X, Y), d-Cl(X, Y) = 1-Cl(X, Y) A 2-CI(X, Y), d-7-l(X, Y) = 1-7-/(X, Y) A 2-7-/(X, Y). Following [31], the bitopological cube is the product
::(x) =
I-[ (I, ::f(x)
where .7"f(X) is the class of all functions t f " (X, 71, T2) --~ (I, Wl, w~), f c d-C(X, I).
Invariants of d-homeomorphisms are called bitopological properties. Thus a bitopological property P belongs to a BS (X, T1, w2) if and only if P belongs to every BS, that is, d-homeomorphic to (X, Wl, T2). Since the object of the theory of BS's is to study bitopological properties, from the bitopological viewpoint two d-homeomorphic BS's can be regarded as one and the same object. In our further discussion we shall abbreviate "lower (upper) semicontinuous"
to 1.(u.)s.c. D e f i n i t i o n 0.1.3. A function f : (X, 7-1,7-2) ---+ (I,O./)is said to be (i,j)-l.u.s.c. if f is i-l.s.c, and j-u.s.c.
0.1. Symbols and Notations. Basic Concepts of Bitopology
Proposition
and only if f :
0.1.4. A function f " (X, rl, r 2 ) ~
!
5
!
(I,~v~,~j) is d-continuous if
( X , T i , T 2 ) --~ ( 1 , ~ ' ) i8 ( i , j ) - l . u . 8. C.
Proof. We shall consider only the case with i = 1 and j = 2 because the other case is proved similarly. Let f " (X, ri, r2) --~ (I, w i, w~) be d-continuous (i.e, 1- and 2-continuous). Then for every 1-open set (a, 1], a c [0, 1), the preimage f - l ( a , 1] c TI and for every 2-open set [0, a), a c (0, 1], the preimage f - l [ 0 , a) E r2. Thus by [79, 12.7.2], f : ( X , T1,T2) ---+ (/,CO')is (1,2)-I.u.s.c. Conversely, let f : (X, rl,r2) ~ (I, co') be (1,2)-l.u.s.c. Then for every set (a, 1], a ~ [0, 1), the preimage f - l ( a , 1] E ~-1, and for every set [0, a), a < (0, 1], the preimage f - l [ 0 , a) c ~-2 since f is (1, 2)-l.u.s.c. Recalling that co~ - {~, I} tO {(a, 1] : a c (0,1)} and co~ = { ~ , I } t 0 { [ 0 , a) : a c (0,1)}, we conclude that f " (X, rl, r2) ~ (I, w~, w~) is d-continuous. [--] Adapting Definition 2.1 from [166] to our terminology, we come to D e f i n i t i o n 0.1.5. Let (X, rl, r2) be a BS and A, B be subsets of X. Then A is (i,j)-completely separated from B if there is an (i,j)-l.u.s.c. function f : (X, rl, r2) -+ (I, co') such that f ( A ) = 0 and f ( B ) = 1. Since f : (X, rl,r2) --+ (I, c0') is (i,j)-l.u.s.c. -: ;- (1 - f ) :
( X , T1, T2) +
-' :-
(I, co t) is (j, i)-I.u.s.c.,
it clearly follows that A is (i, j)-completely separated from B <---> -' ;- B is (j, /)-completely separated from A. We next introduce various properties of BS's which describe how the points a n d / o r / - c l o s e d sets of a BS are separated by means of disjoint/-open and j-open sets. These definitions are necessary to study the influence of pairwise separation axioms on bitopology. Note that in what follows the letter "W" abbreviates the word "weakly". D e f i n i t i o n 0.1.6. Let (X, rl, r2) be a BS. Then (1) (X, rl, r2) is MN-p-T0 (i.e., p-To in the sense of Murdeschwar and Naimpally) if for every pair of distinct points, there exists a 1- or a 2-neighborhood of one point not containing the other [189]. (2) (X, 7-1, r2) is MN-p-R0 if for every set U E 7-1 \ {l~} it follows from x ~ U that r2 cl{x} C U and for every set V E r2 \ {~} it follows from y c V that rl cl{y} C V [189]. (3) (X, r~, r2) is MN-p-T1 if for every pair of distinct points x, y c X, there exists a 1- or a 2-neighborhood of x not containing y [189]. (4) (X, rl, r2) is S-p-T1 (i.e., W-p-T1 in the sense of Swart) if for every pair of distinct points at least one point has a 1-neighborhood not containing the other, while the second point has a 2-neighborhood not containing the first [248]. (5) (X, rl, r2) is R-p-T1 (i.e., p-T1 in the sense of Reilly) if it is d-T1 [215].
6
O. Preliminaries
(6)
(X,T1,T2) is R-p-R1 iffor each pair of points x, y E X such that x E ~-~cl{y} there are an /-open set U and a j-open set V such that x E U, y c V and U N V - 2~ [217]. (7) (X, T1,7-2) is MN-p-R1 if for every pair of points x, y E X, T1 e l { z } r r2 cl{y} implies that x has a 2-neighborhood and y has a 1-neighborhood which are disjoint [189]. (8) (X, T1, ~-2) is W-p-T2 if for every pair of distinct points x, y c X there exist a 1-open set U and a 2-open set V such that either x c U, y E V or x E V, y E U, and U A V - 2~ [152] (see also [31], [248]). (9) (X, ~-~,~'2) is p-T2, (i.e., p-Hausdorff) if for each pair of distinct points x, y E X there exist a l-open set U and a 2-open set V such that x E U, yEV, andUC~V-~ [151]. (10) (X, ~-~,T2) is (i, j)-semiregular if for each point xEX and each/-open set U, x E U, there exists an /-open set V such that x E V C ~-i int ~-j cl V C
u [236]. (11) (X, T1,72) is (i,j)-almost regular if for each point x c X and each (i, j)-closed domain F (see Definition 1.3.3), x E F, there exist a n / - o p e n set U and a j-open set V such that x E U, F C V and U N V - 2~ [236]. (12) (X, T1, r2) is (i, j)-regular if for each point x E X and each/-closed set F, x g F, there exist a n / - o p e n set U and a j-open set V such that x E U, FCVandUAV-o [151]. (13) (X, T1, T2) is (i, j)-almost completely regular if every (i, j)-closed domain F is (i, j)-completely separated from each point x E F [238]. (14) (X, ~-1,T2) is (i, j)-completely regular if every i-closed set F is (i, j)-completely separated from each point x ~ F [166], [189]. (15) (X, T1,7-2) is (i,j)-seminormal if for every j-closed set F and every (i,j)-open domain U (see Definition 1.3.3), F C U, there exists an /-open set V such that F c V c 7-i int ~-j cl V c U [238]. (16) (X, T1, T2) is (i, j)-almost normal if for every pair of disjoint sets A, B in X, where A is j-closed and B is an (i,j)-closed domain, there exist an /-open set U and a j-open set V such that A c U, B c V and U A V [238]. (17) (X, ~-1,~-2) is p-normal if for every pair of disjoint sets A, B in X, where A is 1-closed and B is 2-closed, there exist a 2-open set U and a 1-open set V s u c h t h a t A c U , BcVandUAV-~ [151]. (18) (X, ~-1,T2) is (i,j)-D1 if every j-closed set F has a countable/-base for /-open sets containing F, that is, for every set F E co Tj there exists a countable family {V~}n~=I C ~-i such that for each/-open set U, F c U, there is n E N such that F C Vn c U [180]. Though Definition 0.1.6 provides a wide class of axioms of separation of BS's, it can in fact be expanded further. For the purpose of abbreviating the implications below, obtained on the strength of [189] and Definition 0.1.6, and also the implications in Remark 0.1.9, instead of writing spaces, we shall indicate only the corresponding axioms of separation.
0.1. Symbols and Notations. Basic Concepts of Bitopology
7
p -T4 r (p-norm.AR -p -T~) ::~ p -T389r (p -comp.reg.AR-p-T1) ::~ p -T3 r (p -reg.AR -p -771) :=> R-p-R1
--,
R-p-T1 A MN-p-R1
p-T2 W -p -T~
S-p -T1
d-To
MN-p-T1
MN -p - R0
MN -p-T0 P r o p o s i t i o n 0.1.7. The statements below hold for a BS (X, T1, T2): (1) (X, T1, T2) is (i,j)-almost regular if and only if for each point x c X and each (i,j)-open domain U with x c U, there exists an i-open set V such that x E V C rj clV c U. (2) (X, T1,T2) is (i,j)-regular if and only if for each point x c X and each i-open set U with x c U, there exists an i-open set V such that x c V c 7 j c l V C U. (3) (X, T1,T2) is (i,j)-regular if and only if it is (i,j)-semiregular and (i, j)-almost regular [236]. (4) (X, 7-~,T2) is p-normal if and only if for each 2-closed (1-closed) set F and each 1-open (2-open) set U with F C U, there exists a 1-open (2-open) set V such that F c V c ~-2 el V c U (F c V c 3-1 el V c U). (5) (X, T1, T2) is p-normal if and only if it is p-seminormal and p-almost normal [238]. Moreover, for a BS (X, T1 < T2), we have: (6) (X, 7-1,7-2) is d-reg. ~ ( X , 7-1,72) is 1-reg. ~ ( X , T1, 7-2) i8 (1, 2)-reg.
(x, ~1, ~)
i~ 2 - ~ g . . -
(x, ~, ~) i~ (2,1)-~g.
(7) (X, 7-1, T2) is (1, 2)-completely ((2, 1)-completely) regular if for each point x c X and every 1-closed (2-closed) set F with x-~ F, there exist a 1-1.s.c. a~d 2 - ~ o ~ t i ~ o ~ (1-~.~.~. a~d 2 - ~ o ~ t i ~ o ~ ) / ~ t i o ~ f : (X, ~1, ~2) (I, oa') such that f (x) = 1 and f (F) = O. Therefore (X, rl, T2) is d-compl, reg. ~
(X, rl, T2) is 1-compl. reg. ~
(X, ~-1,72) is 2-compl. reg. ~
(X, ~-1, 72) is (2, 1)-compl. reg.
II
(X, ~-1,T2) is (1, 2)-compl. reg.
C o r o l l a r y 0.1.8. In a p-normal BS (X, T1, T2), for every pair of disjoint sets A , B , where A is l-closed and B is 2-closed, there exist a 2-open set U and a l-open set V such that A c U, B c V and 7-1 C1 U N 7-2 c1 V ~- ~ . Remark
0.1.9. Following [228], W-p-T3 ~
(p-regularity A S-p-T1),
W -p-T3~1 ~, , (p-complete regularity A S-p-T1),
8
0. Preliminaries
W-p-T4 z--> (p-normality A S-p-T1). Moreover, by [205], p-complete regularity implies MN-p-R0 and, hence, for a p-completely regular BS,
W-p-T31 ,z---5, W-p-T3 ,z---5, W-p-T2 ,z----5, S-p-T1 ,z--> MN-p-T0 [205]. D e f i n i t i o n 0.1.10. A BS is said to be hereditarily p-normal if every one of its BsS is p-normal [84]. The class of such spaces will play an essential role in Chapter III dealing with the dimension theory of BS's. D e f i n i t i o n 0.1.11. A BS (X, 7"1,7"2) is said to be (i,j)-perfectly normal if it is p-normal and co 7"i c j-G5(X) [202]. One can easily see that the BS (R, CO1 < CO) is (1, 2)-perfectly normal, but it is not (2, 1)-perfectly normal. By Theorem 2.5 in [202] every p-perfectly normal BS is p-completely normal (see Theorem 0.2.2 below) and so it is hereditarily p-normal. P r o p o s i t i o n 0.1.12. If a BS (X, 7"1,7"2) is p-regular and d-second countable, then it is p-perfectly normal.
Proof. By Lemma 3.2 in [151] and Definition 0.1.11, it suffices to prove that 7-i c j - ~ ( X ) . Let U c 7-i \ {2~} be any set and Bi be a countable/-base. Then for each point x E U there is a set U(x) E 13~ such that x E U(x) c 7-j cl U(x) c U so that U = U 7-j cl U(x). The latter union is countable because all sets U(x) xEU
belong to Bi.
71
C o r o l l a r y 0.1.13. A R~v-T1 and d-second countable BS is p-perfectly normal if and only if it is p-normal.
Proof. Since every R-p-T1 and p-normal BS is p-regular, the proof is an immediate consequence of Proposition 0.1.12 and Theorem 0.2.2 in conjunction with its Corollary 0.2.3. [-1 Using the notion of J. C. Oxtoby [198], we now define (i,j)-quasi regular BS's which play an important role in the formulation and treatment of (i, j)pseudocomplete BS's in Chapter IV. D e f i n i t i o n 0 . 1 . 1 4 . A BS (X, 7l,T2) is (i,j)-quasi regular if for every set U E 7-i \ {2~} there exists a set V E 7-~ \ {~} such that 7-j cl V c U.
It is obvious that every (i,j)-regular BS is (i,j)-quasi regular, but not conversely. Further, if 7-2 C 71 for (X, ~-1,7-2), then (X, 7-1,v2) is (1, 2)-quasi regular if and only if (X, T1,72) is (vl, 7-2)-regular in the sense of [260], in which some interesting examples of such spaces are given. Moreover, for a BS (X, 7-1 < 7-2), we have (X, T1,3-2) is d-quasi reg. ~
(X, Vl, 3-2) is 1-quasi reg. ===, (X, T1,3-2) is (1, 2)-quasi reg.
(X, T1,3-2) is 2-quasi reg. r
(X, 3-1,3-2) is (2, 1)-quasi reg.
0.1. Symbols and Notations. Basic Concepts of Bitopology
9
P r o p o s i t i o n 0.1.15. The conditions below are equivalent for a BS (X, 71,72):
(1) (x,
(i,
(2) For every set UE7~\{2~} there exists a (j,i)-closed domain F such that FcU. (3) The topology 7.~ has a pseudobase (see Definition 2.1.1) consisting of j-closed sets. C o r o l l a r y 0.1.16. The following conditions are satisfied for a BS (X, 71, T2): (1) If (X, 7.1,7-2) is (i,j)-quasi regular and Y c 7-i \ {~} U i-D(X) (see Definition 1.1.9 for j - i ) , then (Y, 7-;, T~) is also (i,j)-quasi regular, and so (2) /f (X, 7.1,7-2) is p-q~tasi regular and Y C ((T 1 if)7-2) \ {~})[-J (1-T)(X)N 2-D(X)), then (Y, 7-{, 7-~) is also p-quasi regular.
Moreover, for a BS (X, 7-1 < 7-2), we have (3) If (X, T1, 7-~.) is p-quasi regular and Y c 7.1 U 2-D(X), then (Y, 7.[, 7.~) is also p-quasi regular. D e f i n i t i o n 0.1.17. Let (X, 7-1,7-2) be a BS. Then (1) (X, T1,T2)is (i,j)-(countably)-paracompact if each (countable)/-open covering of X has a n / - o p e n refinement which is j-locally finite [209]. (2) (X, 7-1,7-2) is (i, j)-locally LindelSf if each point x c X has a j-neighborhood U(x) which is i-LindelSf [210]. (3) (X, 7-1,T2) is (i, j)-paraLindelSf if each/-open covering of X has an/-open countable refinement which is j-locally finite [210]. (4) (X, ~-1,7.2) is p-LindelSf if every proper 1-closed set is 2-LindelSf and every proper 2-closed set is 1-Lindelbf [238]. D e f i n i t i o n 0.1.18. Let (X, 7"1,7-2) be a BS. Then (1) (X, 7-1,7-2) is p-connected if X cannot be expressed as a union of two disjoint sets A and B such that A c 7-1 \ {~}, S C T2 \ {~} [204] (see also [17], [71], [187]). (2) (X, 7-1,7-2) is weakly totally disconnected if for each pair of distinct points there exists a disconnection X = AIB (i.e., X = A U B, where A c 7-1 \ { e } , B C 7-2 \ {~} and A N B = 2~) such that one point belongs to A, the other to B, and the roles of the points need not be interchangeable [248]. (3) (X, 7-1,7-2) is (i, j)-extremally disconnected if rj cl U = 7-i int 7j el U for every set U E Ti or, equivalently, if 7-j el riint A = ri int Tj cl 7-i int A for every subset A c X [12]. It is obvious that for a BS (X, ~-1 < T2) the following implications hold: (X, 71, T2) is d-connected > (X, T1, T2) is 2-connected ~ (X, T1,72) is p-connected ==> (X, 71, T2) is 1-connected.
10
0. P r e l i m i n a r i e s
Using (1) of Lemma 0.2.1 it is not difficult to see that the following three conditions are equivalent: (X, T1,7-2) is (1, 2)-extremally disconnected, (X, 7-1,T2) is (2, 1)-extremally disconnected, and (X, 71,7-2) is p-extremally disconnected. Therefore, later we shall consider only p-extremally disconnected BS's. In contrast to the topological case, the notion of bitopological compactness depending on special coverings of BS's has not been defined uniquely. Many approaches to this notion, in particular the ones given below, are available (see, for example, [113], [248], [72], [88]). D e f i n i t i o n 0.1.19. Let (X, 7-1,7-2) be a BS. Then
(1) (X, 71,7-2) is compact in the sense of Fletcher, Hoyle, and Patty (briefly, FHP-compact) if every p-open covering// = {Us }~cs of X, that is, a f a m i l y / / = {Us}~cs such that // c T~ U7-2, X = U us a n d / / N w ~ s6S
contains a nonempty set, has a finite subcovering [113]. (2) (X, T1,7-2) is quasi compact if the TS (X, sup(7-1,7-2)) is compact [184] (see also [248] and [72]). It is obvious that quite a variety of notions of bitopological compactness gives rise to an even greater variety of notions of bitopological local compactness. How these notions are related to one another is discussed schematically in [184]. Below we recall some of them needed for our discussion. D e f i n i t i o n 0 . 1 . 2 0 . Let (X, Wl, 7-2) be a BS. Then
(1) (X, 7-1,7-2) is (i,j)-locally quasi compact (briefly, (i,j)-lqc) if each point x E Z has an /-neighborhood U(x) such that 7-jcl U(x) is quasi compact [183]. (2) (X, 7-1,7-2) is (i,j)-locally compact in Stoltenberg's sense (briefly, (i,j)-Slc) if each point x E X has an /-neighborhood U(x) such that 7-j cl U(x) is j-compact [245]. (3) (X, 7-1,7-2) is (i,j)-locally compact in Reilly's sense (briefly, (i,j)-Rlc) if each point x E X has an /-neighborhood U(x) such that 7-j cl U(x) is FHP-compact [214]. (4) (X, 7-1,7-2) is (i,j)-locally (countably) compact in Raghavan and Reilly's sense (briefly, (i, j)-RRlc ((i, j)-RRlcc)) if each point x 9 X has a j-neighborhood U(x) which is i-(countably) compact [209]. (5) (X, 7-1,7-2) is (i,j)-locally compact in Birsan's sense (briefly, (i,j)-Blc) if each point x 9 X has an/-neighborhood which is j-compact [30]. Following [184], (i, j)-Rlc .r
(i,j)-lqc --->. (i, j)-Slc ~
(i,j)-Blc ~
(j, i)-RRlc.
In [190] L. Nachbin observed that if the symmetry axiom for a uniform space (X,//) is omitted, the resulting "quasi uniform" space defines, in a natural manner, two topologies 7-1 and 7-2 on the underlying set X (see also D. Tamari [250]), that is, {U(x) : U 9 b/} is the filter of all neighborhoods of a point x E X for the topology 7-1 = 7-(//) and { V - l ( x ) : V -1 9 //-1} is the filter of all neighborhoods of the point x for the topology 7-2 = 7-(//-1), w h e r e / / - 1 = {U-1 : U 9 b/}. Thus for
0.2. I n t e r n a l C h a r a c t e r i z a t i o n of Pairwise C o m p l e t e R e g u l a r i t y
11
every quasi uniform space there is an associated BS. On the other hand, according to [166], a BS (X, T1, ~-2) is quasi uniformizable if there exists a quasi uniformity b / o n X such t h a t T 1 = 7-(~/[) and T2 = ~-(b/-1). In particular, it is shown in [166] t h a t a BS (X, T1, T2) is quasi uniformizable if and only if it is p-completely regular.
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise C o m p l e t e R e g u l a r i t y We begin this section by proving a few elementary facts t h a t will be frequently used in our further investigation. L e m m a 0.2.1. For a BS (X, 7-1, r2), we have (1)
riint F = ri int rj cl r i i n t F for every set F c co 7-j so that ri cl U = ri cl rj int ri cl U for every set U G ~-j
or, equivalently, ri int rj cl A = r i i n t ry cl r~ int ry cl A
so that ri cl rj int A = ri cl rj int ri cl rj int A for every set A c X.
Moreover, for a BS (X, rl < r2), we also have (2) ri cl rj cl A = rl cl A and so ri int rj int A = r l i n t A for every set A c X . Proof. (1) Clearly, it suffices to prove only one of the four equalities, for example, the third one. It is obvious t h a t ri cl rj int ri cl 7-j int A c ri cl rj int A. Conversely, if x c r~ cl rj int A, then U ( x ) • rj int A # ~ for e v e r y / - o p e n neighborhood U(x). Therefore U(x) N rj int ri cl 7-j int A # ~ as rj int A c rj int ri cl Tj int A, and since U(x) E T~ is arbitrary, we obtain x E Ti cl Tj int Ti cl Tj int A. (2) Let us prove the first equality. It is evident t h a t r2 cl rl cl A = rl cl A and rl cl A C rl cl r2 cl A. If x c T1 cl T2 cl A is any point, then for every 1-open neighborhood U(x), we have U(x) NT2 c l d # 2~. Hence V ( x ) ~ d # 2~ because U(x) e 7"1 C 7"2. Since U(x) e T1 is arbitrary, we have x c rl cl A. [2]
T h e o r e m 0.2.2. A BS (X, 71,'/-2) i8 hereditarily p-normal if and only if it is p-completely normal in the sense of Patty [202], that is, if and only if whenever A and B are subsets of X such that (T1 el A N B) U (A N T2 cl B) = 2~, there exist a 2-open set U and a 1-open set V which are disjoint and for which A C U and BcV. Proof. We begin by assuming t h a t (X, "rl, "r2) is hereditarily p-normal and A, B are the subsets of X satisfying the condition (~-~cl A A B) U (A N T2 cl B) = ~. If Y = X \ (T1 cl A ~ T2 cl B), then A C Y, B C Y. By virtue of the hereditary p-normality of (X, T1,7-2) the BsS (Y, T~,7-~) is p-normal and, therefore, for the
12
0. Preliminaries
disjoint 1-closed set A' - 7-1 c1 A N Y and 2-closed set B ' - 7-2 cl B n Y in (Y, 7-~, 7-~) there exist disjoint sets U' c ~-~ and V' c 7-{ such that A' c U' and B ' c V'. Let g l C 7-2, V1 E 7-1 and U1 n Y = U', V1 n Y = V'. Then A c Y n 7-1 cl A = A' c U1,
B C Y n 7-2 cl B = B' c V1.
Since Y -- X \ (T1 c l A n w2 clB) = (X \ T 1 c l A ) u (X \ w2 c l B ) , we have
A c X \ 7.2cl B C 7.2 and B C X \ wl cl A 6 7.1. It is not difficult to verify that U--UIN(X\T2c1B)
CT2 and V = V I N ( X \ 7 - 1 c 1 A )
cT1
are the desired disjoint sets containing A and B respectively. Conversely, let (X, 7.1,7.2) be a p-completely normal BS and (Y, 7~, 7.~) be its any BsS. If ' B E c o T2' and A a B - ~, A c co T1, then (T~ cl A a B) u (A n 7.~ cl B) : ~. Therefore (~-1 cl A n B) U (A N 7.2 cl B) = and, by condition, there exist a 2-0pen set U and l-open set V in (X, 7.1, T2) which are disjoint and for w h i c h A c U a n d B c V. Hence the sets U' = U N Y and V' - V N Y are respectively 2-0pen and 1-0pen in (Y, 7.~, 7.~) which are disjoint, A c U' and B c V'. Thus (Y, 7.~, v~) is p-normal and, consequently, (X, T1,7"2) is hereditarily p-normal. D C o r o l l a r y 0.2.3. Every hereditarily p-normal BS is p-normal. The remainder of this section deals with an internal characterization of p-completely regular BS's. The main result (Theorem 0.2.5) was originally formulated in [83]. A double family, t h a t is, a pair of families Z = {Z1,Z2}, where Zi is an /-closed base of a BS (X, 7.1,7.2), is called a d-closed base and co Z = {co Zi, co Z2 }, where co Zi is a n / - o p e n base, conjugate to Zi, is called a d-open base, conjugate to Z = { Z 1 , Z 2 } . D e f i n i t i o n 0.2.4. A d-closed base Z = {Z1, Z2 } of a BS (X, 7.1,7.2) is said to be p-normal if the following conditions are satisfied: (1) For every point x c X and its any neighborhood U(x) E co ~1 (U(x) E co Z2), there exists a set A c Z2 (A E Z1) such that x c A c U(x). (2) If A E Z1, B E Z2 and A N B = z , then there exist U c co Z2, V E co Z1 such that A c U, B C V and U N V = z . T h e o r e m 0.2.5. A BS (X, 71, T2) is p-completely regular if and only if it possesses at least one p-normal base. To prove this theorem, we have to formulate a few auxiliary statements.
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise Complete Regularity
13
D e f i n i t i o n 0.2.6. Let (X, T1, T2) be a BS. Then the double family A = {A1,A~}, where Ai = {(F, U ( F ) ) : F c coT~ and U(F) is its j-neighborhood}, is said to be p-dense if for every pair (F, U(F)) c A~ there exist a j-neighborhood V(F) and an/-closed set (I) satisfying the following conditions:
(1)
d v(F) c 9 c
(2) (F, V(F)), (q~, U(F)) e A~. L e m m a 0.2.7. Let Z - { Z I , Z 2 } be a p-normal base of a BS (X, T1, T2). Then the double family , 4 - {Al,fl~2}, w h e r e A i - {(A, U(A)) " A c Z~, U(A) c co Zj }, is p-dense.
Proof. Let (A, U(A)) c A~. Consider the set B = X \ U(A) c Zy. It is obvious that A N B = o and by (2) of Definition 0.2.4, there exist disjoint neighborhoods V(A) e c o z y , V ( B ) e cozy. Assume that q~ = X \ V ( B ) e Zi. The equality V(A) N V ( B ) = ~ implies V(A) c 9 and so ~-~cl V(A) c q). On the other hand, (X \ U(A)) ~ q~ - ( X \ U(A)) c~ ( X \ V ( B ) ) - B n ( X \ V ( B ) ) - ~, that is (I) c U(A). Hence AC'qclV(A)
c(FcU(A)
and V(A) c coZj,
q~ c Zi D
imply (A, V(A)), (~o, U(A)) E A~.
L e m m a 0.2.8. Let A and B be disjoint, respectively, i-closed and j-closed subsets of a BS (X, T1, %). If for the set A there exists a family of j-neighborhoods {U~}, enumerated by all dyadic rational numbers {r " 0 < r < 1} in a manner such that U1 - X \ B and rl < r2 implies ~-~cl UF1 C Vr2 , then A is (i,j)-completely separated from B.
Proof. We shall use a generalization of Urysohn's procedure. Let lit = ~ if t < 0, and Ut = X if t > 1. Besides, we shall assume t h a t to each t c [0, 1], which is not a dyadic rational number, there corresponds the j-open set lit = U u~. It ~
r
Hence by condition, ~-i el Utl C Ti cl ST,1 C Sf2 C gt2. Next, let z C X be any point. Then the point z together with the family {Ut : t c R} defines the section of the set R as follows: a real number t belongs to the lower class A x of this section if z-c Ut and t belongs to the upper class B x if z c Ut. Clearly, this section defines a real number ~x. If f : (X, rl, r2) ~ (I, co') is defined as f ( x ) = ~x, then from the definition of this function it immediately follows that f ( A ) = 0 and Z(B) = 1. Thus it remains to prove t h a t f is (i, j)-l.u.s.c. Indeed, assume t h a t z E X is any point and t c R is any real number. If f (x) > t, then t c A x so t h a t z-~ Ut. Since f (x) > I(x)+t 2 = t', where ~'i cl Ut C Ut, and x-4 Ut,, we obtain z E X \ ri cl lit = U(x). On the other hand, if f ( z ) < t,
14
0. P r e l i m i n a r i e s
then t c B z so that x c Ut - V(x). Thus f is (i,j)-l.u.s.c. since U(x) 9 ~-i and V(x) c D L e m m a 0.2.9. Let (A,U(A)) 9 Ai, where A {A1,A2} is any p-dense family of a BS (X, T1, T2). Then A is (i,j)-completely separated from X \ U(A).
Proof. Let (A, U(A)) c Ai. Clearly, it suffices to establish for A the existence of a family of j-neighborhoods {U~} enumerated by all dyadic rational numbers {r 9 0 < r < 1}, which satisfies the conditions of Lemma 0.2.8. Let U1 - U(A). Since A - {A1,A2} is p-dense, there exist a j-open set U0 and an/-closed set (I)0 such that A c U0 c T~ clU0 c ~0 c U~. By the same reasoning as above, there are a j-open set U89 and an/-closed set ~ such that ~o c U!2 c w~clU!2 c ~12 c U1. Thus for each dyadic rational number r - ~ , where 0 _< r _< 1, we can find by induction a j-open set U ~2 n and an/-closed set ~__~ such that 2n (I) _~7
C
C
U 2p-l- 1 2n+ 1
7"i
cl U 2p+ 1 C (~ 2p+ 1 C 2"n+ 1
2'n+ 1
U
2 P+nl ~ "
~-]
Note that following [166] any /-zero set in a BS (X, TI,T2) is of the form f ( x ) - O, f is (i,j)-l.u.s.c. and f _> 0}. This definition immediately implies that a n / - z e r o set is/-closed. Now we can proceed to prove our main result, that is, Theorem 0.2.5. Necessity. Let (X, TI, ~'2) be a p-completely regular BS in the sense of (14) of Definition 0.1.6. Also, let Z - {Z1,Z2}, where Zi is the family of all i-sero sets. By [166, Proposition 2.9], Z - {Zx, ~-2} is a d-closed base of (X, T1, T2). Let us show that conditions (1) and (2) of Definition 0.2.4 are satisfied. (1) Let x 9 X be any point and g(x) 9 co Zj be its any neighborhood. Since (X, Wl, T2) is p-completely regular, the point x is (i,j)-completely separated from the j-closed set X \ U(x), that is, there exists an (i, j)-l.u.s.c, function
{x C X "
f " (X, TI,~'2) --, (I,a/) such that f ( x ) -
0 and ( f ( X \ U ( x ) ) - 1.
Assume that A - f - l ( 0 ) . Clearly, A 9 Z~ and x 9 A c U(x). (2) Let A 9 Z1, B 9 Z2 and A n B - ~. Then there exist a (1,2)-l.u.s.c. function r _> 0 and a (2,1)-l.u.s.c. function ~ _> 0 such that A - r B - ~-1(0). By [166, Proposition 2.8], there exists a (1,2)-l.u.s.c. function h" (X, T1, w2) ~ (I, co') such that h(A) - 0 and h(B) - 1. Let 0 hi(x)
--
3
1 {h(x)-~}
if 0 _< h(x) <_ -~, 1 if < h(x)<_1
and 0 gl(x)
-
-
1 ~ _< h(x)<_ 1, 1 if O < h ( x ) < if
/ 1 \ 3(-z - h(x)) \j
/
0.2. Internal C h a r a c t e r i z a t i o n of Pairwise Complete Regularity
15
It is obvious t h a t 0 < hi(x) < 1, 0 < gl(x) <_ 1 for each point x E X, hi is (1, 2)-l.u.s.c. and gl is (2, 1)-l.u.s.c. Let Ai - h11(0), B1 - g~-l(0). Then A1 e Z1, - U(B) e B1 E Z2. Moreover, 0 < h(A1) < 1 and h ( B ) - 1 g i v e B c X \ A 1 co Z1, while ~1 <_ h(B1) <_ 1 and h ( d ) - 0 give A c X \ B1 - V ( d ) e co Z2 It is easy to see t h a t A1UB1 - h11(0)Ug11(0) - X so t h a t U ( A ) N U ( B ) - ~. Sufficiency. Let Z - {Z1, Z2} be any p-normal base of a BS (X, ~-1, T2). Consider the double family . 4 - {A1,A2}, where A~ - {(A, U ( A ) ) " A e Z~, U(A) e c o z y } . Then by L e m m a 0.2.7, A - {A1,A2} is p-dense. If x c X, F E coT~ and x ~ F , then there exists a set A E Zi such t h a t F c A and x ~ A since Zi is the /-closed base. It is obvious t h a t U(x) - X \ A c co Zi. Since Z - {Z1, Z2} is the p-normal base, there exists a set B c Zj such t h a t x E B c U(x). Hence A N B and by L e m m a 0.2.9 there exists an (i,j)-l.u.s.c. function f 9 (X, T1,T2) ~ (I,w') suchthat f(A)-Oandf(B)1. B u t x e B a n d F C d . Thus f ( x ) land I(F) -0. Further, we have
Proposition 0.2.10. A d-closed base Z = {Z1, Z2} of a BS (X, p-normal if and only if the following conditions are satisfied:
71,72)
is
(1) For every point x E X and every set A c Z1 (A c Z2), x-~ A, there exist U c c o Z 1 , V c co Z2 (U c co Z2, V E co Z1) such that x c U, A c V, andUNV=~. (2) I r A c Z1, B c Z2 and A N B = ~, then there exist U c coZ2, V c coZ1 such that A c U, B c V, and U n V = ~.
CHAPTER I
Different Families of Sets in Bitopological Spaces
We begin the study of different families of subsets of a BS (X, T1,7"2), using the bitopological modifications of the fundamental topological notions of a dense set; a boundary (also called codense) set; a derived set; a dense in itself set; a perfect set and a scattered set belonging to G. Cantor [54], [55], [57], a nowhere dense set defined by P. du Bois-Reymond [34], first and second category sets introduced by R. Baire [20], and the boundary of a set studied by G. Cantor [56] and C. Jordan [148]. The obtained modifications - - the (i, j)-boundaries and (i, j)category n o t i o n s - play an essential role, the former in constructing the dimension theory for BS's in Chapter III and the latter in establishing the principal Baire-like properties of BS's in Chapter IV. This chapter also deals with the bitopological analogues of open and closed domains of K. Kuratowski [160], [162], and semiopen and semiclosed sets of N. Levine [170]. Moreover, the notions of (i, j)-open domains and that of (j, /)-closed domains make it possible to define the families of (i, j)-semiopen domains which equal the families of (j, i)-semiclosed domains. Their corresponding topological families are new for topology. Since the obtained families, including their two topological versions, are not comparable by inclusion, which means that none of them is contained in the other even when one of the topologies is finer than the other, we introduce the "quotient"-families leading to the solution of the problem of comparability. The p-dense in itself, (i,j)-perfect and p-scattered sets are also investigated [95] and a bitopological analogue of the well-known Cantor-Bendixson's theorem is proved. Furthermore, we introduce and study three special operators on 2 X, which give an exact determination of degrees of nearness of the four boundaries of any subset of a BS (X, 7-1 < 72). Based on our earlier studies, we end the chapter by defining and investigating the so-called relative properties, namely the properties of subsets of BsS's of BS's. Among the obtained results the most important one shows that for a BS (X, T1 < 72) every non-empty 1-open set has the same (1, 2)-category both in itself and in the whole space, while for a non-empty 2-open set the corresponding (2, 1)-categories may not coincide. The chapter provides a few simple and illustrative examples.
16
1.1. (i,j)-Nowhere Dense Sets and ( i , j ) - C a t e g o r y Notions
17
1.1. (i, j)-Nowhere Dense Sets and (i, j)-Category Notions The notion of an (i, j)-nowhere dense set is one of the frequently used notions in this work. It will underlie the definitions and investigation of the Baire-like properties for BS's some of which are defined in [173].
Definition 1.1.1. A subset A of a BS (X, r l,r2) is termed (i,j)-nowhere dense (or (i,j)-rare) in X if rj cl A contains no n o n e m p t y / - o p e n set, that is, to say if r i i n t rj cl A = ~. The families of all (i,j)-nowhere dense subsets of X (i,j)-N'D(X). A simple application of the foregoing definition yields
are denoted
by
Proposition 1.1.2. Let (X, rl, r2) be a BS and A c X . Then A e (i,j)-JVD(X) ~
rj clA c r~ cl(X \ rj clA).
Proof. The implication from left to right is obvious. If rj cl A c ri cl(X \ Tj cl A),
then rj cl A N r~ int rj cl A - r, int rj cl A - ~.
Theorem 1.1.3. Let (X, evcfy set U C (T1 n 7"2) \ {~} V n A = ~. Conversely, if for such that V c U and V C~A =
[21
rl,r2) be a BS and A E ( i , j ) - N ' D ( X ) . Then for there is a set V c rj \ {2~} such that V c U and every set U e ri \ {2~}, there is a set V e rj \ {2~} ;~, then A c ( i , j ) - A f D ( X ) .
Proof. First assume that A E ( i , j ) - A f D ( X ) and U c (rl At2) \ {2~} is an arbitrary set. Then U \ rj cl A r 2~ since the contrary means that
2~ - U C~ri el rj int(X \ A) - U c~ (X \ ri int rj el A) - U, which is impossible. Therefore V = U \ rj cl A is the desired set. On the other hand, if A g (i, j ) - N ' D ( X ) , then every set V E rj \ {~}, satisfying the inclusion V c r i i n t rj cl A, is contained in rj cl A, that is, V N rj cl A = V -r 2~. Hence V N A r 2~. [B
Corollary 1.1.4. For a BS (X, T 1 < 7"2) a set A c (1, 2)-2VD(X) if and only if for every set U c 7-1 \ {~}, there is a set V c 7"2 \ {~} 8ugh that V c U and VNA=~. Corollary 1.1.5. The inclusions
(2,1)-ND(x) c 2-:VD(x) n n 1-AfD(X) c (1,2)-AfD(X) are satisfied for a BS (X, rl < r2).
Corollary 1.1.6. For a BS (X, r l < w2) the condition A c (2,1)-AfD(X) implies that for every set U c ~-2 such that rl int U 7~ 2~, there is a set V E rl \ {2~ }
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20
I. Different Families of Sets in Bitopological Spaces
Proof. Let 72 int 7-1 cl A - ~ - Te int T1 cl B. Then 71 cl A E 2-Bd(X) and by (2) of L e m m a 0.2.1, 72 int we el T1 el B -- 7"2 int T1 el B -so t h a t 71 c l B C 2-N'Z)(X). For the topological case, Proposition 1.1.12 gives T1 C1A U T1 cl B E 2-Bd(X). Since T1 cl A U T1 cl B - T1 cl(A U B) c co T1,
we obtain ~-2 int 7-1 cl(A U B) - z ,
t h a t is A U B E (2, 1)-N'~D(X).
V]
D e f i n i t i o n 1.1.16. A family A - {As}sEs of subsets of a BS (X, T1, ~-2) is t e r m e d / - l o c a l l y finite at a j-dense set of points of X if for every set U E ~-i \ {2~} there e x i s t s a s e t V e T j \ { ~ } s u c h t h a t VC Uand {so S" VnAs r Z}is finite. Clearly, if A - {As}sES is/-locally finite at a j-dense set of points of X, then the family j - C 1 A - {~-j cl As}sES is also /-locally finite at a j-dense set of points of X. T h e o r e m 1.1.17. If a family A - {As}sES C (2, 1)-AfZ)(X) is 2-locally finite at a 1-dense set of points of a BS (X, T1% 7"2), then U As c (2, 1)-N'~D(X). sES
Proof. Let A - {As}sES be 2-locally finite at a l-dense set of points of X. Then I-C1A - {~-1clAs}sES must also be 2-locally finite at a l-dense set of points of X so t h a t for every set U E ~-2 \ { z } , there exists a set V c T1 \ { ~ } such t h a t V C U and {sk E S " V N T l c l A s k ~ 2~} is finite. Let {~-lclA~k}~=l be the corresponding finite family. Since 7-1 clAsk C (2, 1)-AfZ?(X) for every k - 1,n, X \ T1 el Ask E T1 N 2-~)(X) for every k - 1, n. One can easily verify t h a t W = n V A ( N ( X \ T1 el Ask)) ~ ~ . Contrary, let W - 2~ so that k=l n n
c
\(
U k=l
k=l
In t h a t case n
n
n
~ T1 int U 7"1 c1 Ask - - T 1 int T1 c1 U Ask C T2 int T1 C1 U Ask' k=l k=l k=l n
which is impossible since, by Corollary 1.1.15,
U Ask c (2, 1)-N'D(X).
Hence
k=l
W c T1\{o}. On the other hand, VC~71 clAs - ~ for s ~ sk a n d s o V c X\T1 clAs for every s r sk. Therefore the set W c V satisfies the inclusion WC
N (X\rlClAs)-X\ sES
UT-lclAs" sES
Hence W E ~-1 \ {2~} implies W C TliIlt ( X \ U T l a l A s ) ~ X \T1c1 U TlClAs C X \T1C1 U As.
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22
I. Different Families of Sets in Bitopological Spaces
Pro@ (1) Using Theorem 2 from [161, p. 78], we obtain {x E X :
T2 el A-~ 1-Bd(X, x) } = 7"1 c17"1 int 7"2 el A.
Thus, it remains to use Proposition 1.1.19. (2) It is clear that { X C A"
A c (1, 9~)-.~:)(X, x) } - A \ 7"1 cl 7"1 int 7"2 cl A c 7"2 cl A \ 7"1 int 7"2 cl A
and 7"1 int(T2 el A \ T1 int 7"2cl A) = n i n t 7"2cl A A (X \ 7"1 el 7"1 int 7"2cl A) = ;~. Therefore 7"2clA \ 7"1 int T2 clA E c07"2 A 1-Bd(X) C (1, 2)-A/D(X).
K]
In particular, if A is (1, 2)-nowhere dense at each of its points, then A = A \ 7"1 cl 7"1 int 7"2cl A so that 7"1 int 7"2 clA = 2~ <---> A e (1, 2)-AfD(X). D e f i n i t i o n 1.1.21. A subset A of a BS (X, 7"1,7"2) is termed (i, j)-somewhere dense in X if A-~ (i, j)-AfD(X), that is, if T~ int 7"j cl A ~ ~. The families of all (i,j)-somewhere dense subsets of X are denoted by Clearly, (i, j ) - S D ( X ) = 2 X \ (i,j)-N'D(X) and hence for the natural BS (R, aJl,W2), we have aJi \ {~} c i-D(R) = 2 R \ i-AfD(R) = i - $ D ( R ) . It is easy to ascertain that the inclusions
(i, j ) - S D ( X ) .
(1,2)-SD(X) c 2-SD(X) A A 1 - $ D ( X ) C (2, 1)-SD(X) hold for a BS (X, T1 < 7"2). In order to use category notions to study bitopological concepts of Baire spaces, we must relate them to the bitopology on a set. D e f i n i t i o n 1.1.22.
A subset A of a BS (X, T1,72) is of (i,j)-first category oo
(also called (i,j)-meager, (i,j)-exhaustible) in X if A -
[.J An, where An e n=l
(i,j)-AfD(X) for every n = 1, oc and A is of (i,j)-second category (also called (i, j)-nonmeager, (i, j)-inexhaustible) in X if it is not of (i, j)-first category in X. A subset A of X is of (i, j)-first category if A is of (i, j)-first category in itself and A is of (i, j)-second category if it is of (i, j)-second category in itself. The families of all sets of (i, j)-first ((i, j)-second) categories in X are denoted by (i, j)- Catg~ (X) ((i, j)- Gatgii (X)), while the statement X c (i, j)- Catg I (X) (X ~ (i, j)-Catgii (X)) is abbreviated to X is of (i, j)-Catg I (X is of (i, j)-Catg II). It is clear that
(i, j)-AfD(X) c (i, j)-datg~ (X) = 2 X \ (i, j)-Catgii (X). R e m a r k 1.1.23. Let T1 = ~ and T2 be respectively the natural and the discrete topology on R. Then n e N implies {n} e (1,2)-AfD(R) and hence N e ( 1 , 2 ) - d a t g i ( R ). On the other hand, if 7~ and T~ are the corresponding
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24
I. Different Families of Sets in Bitopological Spaces oo
n -- 1, oo. Hence F -
U F~ E (i, j)- Cat& (X), and since X \ A
c F, by (1)
n=l
above X \ A E (i, j)- Catg~(X). Thus A E (i, j)- Catgii (X) since by virtue of (1) the contrary means that X is of (i, j)-gatg I. (6) First let {Un}n~176 be a sequence in X, where Un E 7-j a i-D(X) for every oo
n--l,
oo
cxDand n Un--~. T h e n X - X \ n=l
oo
n Un-- U (X\Un),where n=l
n=l
X \ Un c co 7-j n i-13d(X) C (i, j)-Afg)(X) for every n - 1, oc. Therefore X is of (i, j)-Catg I. Conversely, i f X is of (i, j)-gatg I, then by (3), X E j-J~(X)n(i,j)-Cat&(X), oo
i.e. X -
U Fn, where Fn E co 7-j N i-13d(X) for every n - 1, oo. It is clear that n--1
(3o
N(X\Fn)-z,
where {X\Fn }~--1 is a sequence of/-dense j-open subsets of X. IN
n--1
C o r o l l a r y 1.1.25. If (X, 7-1 < 7-2) is of 1-CatgII, then 1-Ga(X) N 2-D(X) C (2, 1)-Catg~i (X).
Proof. By virtue of Proposition 1.5 from [133], X being of 1 - C a t g I I implies 1-Ga(X) n 1-D(X) C 1-Catg~ (X). Hence since 7-1 C 7-2, w e obtain 1-Ga(X) n 2 - D ( X ) C 1 - G a ( X ) n 1 - D ( X ) C 1-Catgi,(X ) C (2,1)-Catgii(X).
D
Further we shall prove some statements connected with Corollary 1.1.25. For the most general result among bitopological results of the type of Proposition 1.5 from [133] reads as follows: if X is of (2, 1)-datg II, then 7-1 C 7-2
2-Gb(X) n 1-D(X) C (1, 2)-Catgii (X) (see (2) of Corollary 2.1.13 below). Nevertheless without resorting to additional assumptions, we have the following weaker statement. Proposition
1.1.26. If (X, 7.1 < 7.2)is of (1,2)-CatgII, then 2-Ga(X) a 1-D(X) C (1, 2)-Cat9~i (X). oo
Proof. Assume that A E 2 - G a ( X ) n 1-/)(X), that is, A -
n An, where An E n=l
r2 n 1-D(X) for every n = 1, oo. Then oo
X \ A - U ( x \ An) E (1, 2)-Catg I (X) n--1
since X \ An E COT2 O 1-Bd(X) C (1, 2)-AfD(X) for every n = 1, oc. If A E (1, 2)-Catgi (X), then by (1) of Theorem 1.1.24 the union A U (X \ A) = X is of (1, 2)-Catg I, which is impossible. D C o r o l l a r y 1.1.27. If a BS (X, Wl < 7.2) is of (1, 2)-datg II, then
1.2. (i, j)-Locally Closed Sets
25
1-Gs(X) N 2-7?(X) C 2-Gs(X) N 2-D(X) N G 1-{75(X) N 1-79(X) C 2 - G a ( X ) n 1-D(X) C (1, 2)-CatgIi (X). Finally, we establish the conditions for sets of (2, 1)-first category in the whole space to become 2-boundaries. Theorem
1.1.28. /f a BS (X, T 1 < 7"2) is 1-compact and (2.1)-quasi regular,
then (2, 1)-Cat&(X) c 2-Bd(X). O<3
Proof. Let A c (2, 1 ) - C a t g i ( X ) be any set, that is, A -
O An, where An n=l
(2, 1)-AfD(X) for each n = 1, oc. Then we must show that V N (X \ A) : / o for each set V c r2 \ {0}. It is evident t h a t if V c r2 \ {0} is an arbitrary set, then V \ rl cl A1 ~ r2 \ { 0 } since V ~ r2 and r2 int rl cl d l = 0. Hence, by condition, there exists a set U1 ~ r2 \ { 0 } such that rl cl U1 C V and 7-1 cl U1 N A1 = 0. Again, there exists a set U2 c r2 \ {0} such that U2 C rl cl U2 C U1 c rl cl U1 and rl cl U2 N A2 = 0 . Following the above reasoning, we can construct a sequence {Un}n~176of non-empty 2-open sets such that rl cl U~ C Un-1 and rl cl UnnAn -- 0 oo
for each n -
1, o0.
But (X, rl < r2) is 1-compact and thus
n r l c l U n =~ 0. n--1
Therefore there exists a point x c X such t h a t x c rl cl Un and x g An for each oo
n-1,
co. T h u s x e V \
U An-V\AandsoVN(X\A)=/=o.
[-1
n=l
1.2.
(i,j)-Locally Closed Sets
D e f i n i t i o n 1.2.1. A subset A of a BS (X, rl, r2) is said to be (i,j)-locally closed at its point x if there exists a set U E ri such t h a t x E U and U O A -
U nrjclA. For every point x E X the families of all sets t h a t are (i,j)-locally closed at the their common point x are denoted by (i, j ) - s x). A subset A of a BS (X, rl, r2) is (i, j)-locally closed if it is (i, j)-locally closed at each of its points. The families of all such subsets of X are denoted by (i, j)-s For a BS (X, rl < r2) the following inclusions hold for every point x c X : 1-/2C(X,x) C (2, 1)-/2C(X,x) n N
< 2-cc(x,
)
and, therefore, 1-s N
c (2, 1)-s N c
Theorem
1.2.2. Let A be a subset of a BS (X, 7-1 ~ 7"2). Then the set
{x c A" A is not (2, 1)-locally closed at a point x} coincides with the set A N r2 el(r1 clA \ A).
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28
I. Different Families of Sets in Bitopological Spaces
so that U n A is j-closed in U. Hence U n A - U n Tj cl A so that A is (i, j)-locally closed at the point x. D Note that the requirement that the BS (X, 7-1,7-2) in Theorem 1.2.5 be (i,j)-regular, is essential. Indeed, if (X, 7-1,7-2) is not (i,j)-regular, then there are a point x 9 X and a neighborhood U(x) 9 7-4 such that 7-j el V ( x ) n (X \ U(x)) r 25 for every/-open neighborhood V(x). From Corollary 1.2.4 it follows that U(x) 9 ( i , j ) - E d ( X ) and thus U(x) 9 (i, j ) - s x). If the condition of Theorem 1.2.5 is fulfilled, then there exists an/-neighborhood W ( x ) such that
n U(x) -
cl ( W ( x ) n U(x))
Let H - 7-4int W(x). Then H n U(x) - E(x) 9 Ti and 7-j clE(x) - 7-j cl(H N U(x)) = 7-j cl (7-4int W ( x ) n U(x)) C 7-j cl (W(x) n U(x)) C U(x), which contradicts the assumption. 1.3. (i, j ) - B o u n d a r i e s . (i, j ) - O p e n D o m a i n s a n d ( i , j ) - C l o s e d D o m a i n s . (i, j ) - S e m i o p e n Sets a n d (i, j ) - S e m i c l o s e d Sets. (i, j ) - S e m i o p e n Domains and (i,j)-Semiclosed Domains Of considerable importance later on is D e f i n i t i o n 1.3.1. For any subset A of a BS (X, 7"1,7"2), the (i, j)-boundaries (i.e., bitopological boundaries) of A are the p-closed sets (i, j)-Fr A - 7-4clA n 7-j cl(X \ A) [84]. The most important properties of these operators are listed in T h e o r e m 1.3.2. For a BS (X, as follows:
7"1,7"2) the
(i,j)-boundaries have the properties
( i , j ) - F r A = B1U B2, where B1 9 j - B d ( X ) and B2 9 i-Bd(X). 7-i cl A = 7j int A U (i, j)- Fr A. Tj int A = A \ (i, j)- Fr A. (i, j ) - F r A = (j, i)-Fr(X \ A). X = Tj int d U (i, j ) - F r A U 7i int(X \ A). (i, j)- Fr T4 cl A U (i, j)- Fr Tj int A c (i, j)- Fr A. A 9 Tj ~ ( i , j ) - F r A = T4clA \ A. A 9 co T4 ~ (i, j)- Fr A = A \ Tj int A. A 9 Tj n co Ti ~ (i, j ) - F r A = ~. (i,j)-Fr A U ( i , j ) - F r B = ( i , j ) - F r ( A U B ) U ( i , j ) - F r ( A N B ) U ( ( i , j ) - F r AN ( j , i ) - F r B ) d ( ( i , j ) - F r B n (j,i)-FrA). (11) If (X1, T} 1) , 7-2(1)) and (X2, w}2) , 7(2)) are BS's with their Catresian product (X, 71, w2) so that X - X1 • X2, 74 - @1) x 7-(2), then the formula
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(i, j)- Fr(A1 • A2) - ((i,j)-FrA1 • T~2) clA2)t2 (7-(1) clA1 • (i,j)-FrA2) holds for every subset A1 • A2 of X , where Ai c Xi.
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34
I. D i f f e r e n t F a m i l i e s of S e t s in B i t o p o l o g i c a l S p a c e s
so that A c (1,2)-OD(X). Hence, taking into account the inclusions
1-OD(X) c (1,2)-OD(X) c
T1
and (2.1)-OD(X) C 2-OD(X),
we find that 7-1 (]
(2, 1)-OD(X) -- (1, 2)-OD(X) N (2, 1)-OD(X) -
= 1-OD(X)fq (2, 1)-OD(X) c 1 - O D ( X ) n 2 - O D ( X ) C
7"1 (-] 2 - O ~ ) ( X ) -
2)-or(x) n 2-Or(X).
:
The rest is obvious since A E (i,j)-OD(X) ~
X \ A E (i,j)-CD(X).
[3
Here note that such a simple argument as T1 c ~-2 is helpful in establishing the inclusions 1-OD(X) C (1,2)-OD(X) and (2, 1)-OD(X) c 2-OD(X). Indeed, A c I-OD(X) implies A
-
T1
int
T1
el A and so A
-
T1
int A.
Hence A G T1
int T2 cl A c
T1
int
T1 c l A -
that is A
A,
-
T1
int ~'2 cl A.
If A c (2, 1)-OD(X), then A - ~-2int T1 clA. Thus A c ~'2 int T2 cl A c ~-2int
T1
el A
--
A so that A c 2-OD(X).
The reverse inclusions in the foregoing corollary are not, generally speaking, correct for a BS (X, T1 < T2). E x a m p l e 1.3.8. Let X and
T1 be as
in Example 1.3.5, 7-2 - {~, {a},{c},{a, c},
{a, b},{a, b, c}, X}. Then {a} E 1 - O D ( X ) c (1,2)-OD(X),
{a} -@2-OD(X)
so that {a} ~ (2, 1)-OD(X). Also, it is clear that {a, b} c (2, 1)-OD(X)
<
2-OD(X),
but {a, b} ~ (1, 2)-OD(X) and, therefore, {a, b} g 1-OZ)(X). E x a m p l e 1.3.9. Let X be as in Example 1.3.5, T 1 - - {~, {a}, {a, b, c},X}, T2 - {Z, {a}, {d}, {a, d}, {b, c}, {a, b, c}, {b, c, d}, X}. Then {a} E (1,2)-OD(X) \ 1-OD(X), {a,d} c 2-OD(X) \ (2, 1)-OD(X). By Example 1.3.9, T1
cl{a} \ {a} = X \ {a} = {b, c, d} -4 2-13d(X)
so that, generally speaking, the difference Tj cl A \ A need not be a n / - b o u n d a r y set for a subset A c ~-i of a BS (X, T1, T2). However, we have P r o p o s i t i o n 1.3.10. The following conditions are satisfied for a BS (X, T1, T2): (1) If A E (i,j)-OD(X), then Tj clA \ A e i-Bd(X).
1.3. (i, j)-Boundaries. (i, j)-Open Domains and ...
(2) If A1, A2 ~ ( i , j ) - 9
A1 c A2 ~
35
then rj clA1 c rj clA2
so that if A1, A2 e (i, j ) - d D ( X ) , then A1 c A2 ~
rj int A1 c rj int A2.
For a BS (X, rl < 72), we also have: (3) U c r2 ~ (U = V \ r 2 c l A , where V c ( 2 , 1 ) - O : D ( X ) a n d A c (1, 2)-N'T)(X)) so that F c coT2 ~ ( F = B U T 2 c l A , where B c ( 2 , 1 ) - r i D ( X ) and A c (1, 2)-AfT)(X)).
Proof. (1) ~-~int(Tj cl A \ A) = Ti int Tj cl d n ~-i int(X \ A) = d A ~-i int(X \ A) = ~. (2) It is obvious that A1 c A2 implies ~-j cl A1 c ~-j cl A2. On the other hand, if ~-j cl A1 c Tj cl A2 (i.e., Tj cl A1 N Tj int(X \ A2) = ~), then
A1 nT-j i n t ( X \ d 2 ) = 2~ (i.e., A1 NTiclTj i n t ( X \ A 2 ) = 2~) since A1 e(i,j)-OTP(X). Hence
A2 c (i,j)-O~D(X)
i m p l i e s n l ffl ( X \ A2) -- z &lid so A1 c A2.
The second part is obvious. (3) Let U c ~-2 and V = 7~ int 7-1 cl U. The difference 7-1 CI V \ U C co 7-2 n 1-gd(X) c (1, 2)-JVZ)(X)
and hence A = V \ U e (1, 2)-AfD(X). Clearly, U = V \ A and r2 cl A = r2 cl(V \ U) c r~ cl V N r2 cl(X \ U) = r~ cl V \ U. Thus
U= VNU=
V \ ( r 2 c l V \ V) c V \ r 2 c l A
c V\A=U,
that is, U = V \ r2 cl A. The reverse implication is obvious. If F c co r~, then X \ F = V \ r 2 c l d , where V c (2, 1)-OD(X) and d c (1, 2)-N"T~(X). Hence F = (X \ V) U w~ clA = B U ~-2 clA with B e (2, 1)-UT?(X).
D
D e f i n i t i o n 1.3.11. A subset A of a BS (X, T1,T2) is called (i,j)-semiopen in X if there exists an /-open set U C X such that U C A c TjclU. The complement in X to an (i, j)-semiopen set is (i, j)- semiclosed in X, that is, a subset B of X is (i,j)-semiclosed in X if there exists an/-closed set F C X such that ~-j int F c B c F [861, [1751. The families of all (i, j)-semiopen ((i, j)-semiclosed) subsets of X are denoted by ( i , j ) - S O ( X ) ( ( i , j ) - $ C ( X ) ) . The theorem below is an immediate consequense of Definition 1.3.11. Theorem
1.3.12. Let (X, T1, ~-2) be a BS. Then
A E ( i , j ) - $ O ( X ) <--> A c Tj cl ~-~int A
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~
"~
~
~
U
38
I. Different Families of Sets in Bitopological Spaces
(4) if ~-j int Bt c C c Bt for some t E T, then C E 13. Then ( i , j ) - S C ( X ) C 13 and, therefore, ( i , j ) - $ C ( X ) is the smallest family of subsets of X satisfying (3) and (4).
Proof. It is clear that by (1) of Corollary 1.3.13 and the first condition in Proposition 1.3.16, the family ( i , j ) - S O ( X ) satisfies (1) and (2). If A E ( i , j ) - S O ( X ) , then by Definition 1.3.11, there is a set U c ~-~ such that U c A c Tj cl U. Since U E ~-i C A, by (2) above A c A. Thus (i, j ) - S O ( X ) c A and (i, j ) - S O ( X ) is the smallest family of subsets of X, satisfying (1) and (2). The rest is obvious. [-1 P r o p o s i t i o n 1.3.19. The following equalities hold for a BS (X, T1, T2) : ~-i = i-Int(i, j ) - S O ( X ) = i-Int(i-SO(X) ).
Proof. If U c T~, then U c ( i , j ) - S O ( X ) . But U = T~intU so that U c i-Int(i,j)-SO(X) and thus ri C i-Int(i,j)-SO(X). The reverse inclusion is obvious by virtue of the definition of the family i-Int(i, j ) - S O ( X ) . Lemma 2 from [170] gives the second equality. D R e m a r k 1.3.20. For a BS (X, 71, r2) any union (intersection) of (i, j)-semiopen ((i, j)-semiclosed) sets is (i,j)-semiopen ((i,j)-semiclosed)[86], [175]. But the intersection (union) of two (i,j)-semiopen ((i,j)-semiclosed) sets may not be (i, j)-semiopen ((i, j)-semiclosed) [175, Example 3]. However, there holds P r o p o s i t i o n 1.3.21. For a BS (X, T1 < T2), we have
(U E 7-1 and A E ( i , j ) - S O ( X ) ) ~
U n A E (i,j)-$O(X)
so that ( F E COT1 and A E ( i , j ) - S C ( X ) ) ---5, F U A E ( i , j ) - S C ( X ) .
Proof. Let U E rl, A E (i, j ) - S O ( X ) and U n A r 2~. For the set A, there is a set Vcrisuchthat VcAcrjclV. Clearly, U N A C ~ i m p l i e s U n r j c l V C ~ s o that U n V r ~ as U E T1 C 7"2. Hence UNV C UNA c UnrjclV
C 7-jcl(UNV),
where U N V E Ti. The second implication follows from the first one. Remark 1.3.20 leads to
[-1
D e f i n i t i o n 1.3.22. Let (X, T1,7"2) be a BS and A c X be any subset. Then
(i,j)-sintA - U {As
As E ( i , j ) - S O ( X ) , As c A for every s c S}
sES
and (i, j ) - s c l A - n sES
Is6], [ 75]
{As
As E (i, j ) - S C ( X ) , A c As for every s E S}
1.3. (i, j ) - B o u n d a r i e s .
(i, j ) - O p e n
Domains
and . . .
39
Thus ( i , j ) - s i n t A are the largest (i,j)-semiopen sets, contained in A and (i, j)- scl A are the smallest (i, j)-semiclosed sets, containing A. Hence
A c ( i , j ) - $ O ( X ) <--> A = ( i , j ) - s i n t A ,
A c (i,j)-Sg(X)
<--> A = (i,j)- sclA
and (i, j)- sint A = X \ (i, j)- scl(X \ A). Moreover, by (1) and ( 2 ) o f Corollary 1.a.la, T~ int A c (i, j)- sint A n i- sint A and
(i,j)- sclA U i- sclA c ~-i clA for every subset A c X. R e m a r k 1.3.23. Following (1) of Corollary 1.3.15, for every subset A of a BS (X, rl < r2), we have: r l i n t A c (1, 2)- sint A c 1- sint A N O O T2intAc 2-sintA c (2,1)-sintA so that (2,1)- sclA C 2- sclA c r2clA N N N 1- sclA C (1,2)- sclA c T1 clA. T h e o r e m 1.3.24. subset A c X "
For a BS (X,
7-1,7-2)
the equivalences below hold for any
(1) A E ( i , j ) - A f D ( X ) ~ (i,j)-sint(j,i)-sclAz. (2) A e i - D ( X ) ~ i-sclA- X ~ (i, j ) - s c l A X.
Proof. (1) Let A c (i,j)-Af~P(X), that is, r i i n t r j c l A - z . Then ( i , j ) - s i n t T j c l A = z . Contrary: (i, j)- sint rj c l d # ;~ and so, there is a set B e (i, j ) - $ O ( X ) \ {;~} such t h a t B C rj clA. By Definition 1.3.11, there is a set V E ri \ { z } such t h a t V C B c r j c l V , and, therefore, V c B c r j c l A . Hence ~ - i i n t r j c l A # ~, which contradicts A c (i,j)-Af~P(X). Since (j,i)- scl A c rj clA, we find t h a t (i, j)- sint(j, i)- scl A - z . Conversely, let (i, j)- slut(j, i)- scl A - ~. Then T~ int(j, i)- scl A - ;~. Assume t h a t F - (j,i)- sclA c ( j , i ) - $ d ( X ) . By Definition 1.3.11, there exists a j-closed set B such that r i i n t B C F c B and, consequently, A c B so t h a t rj cl A c B. Moreover, r i i n t B C r i i n t F - Ti int(j, i)- sclA - ~ and, therefore, Ti int B - z , which eventually gives r i i n t rj cl A - ~. (2) Since (i, j)- scl A U i- scl A C ~-~clA, it is evident t h a t (i, j)- sclA - X or i- scl A - X implies A c i - D ( X ) . Thus it remains to prove t h a t Ti clA - X implies i- scl A - X since the implication ri cl A - X > (i, j)- scl A - X can be proved in a similar manner. Let ri clA - X and i- sclA # X. Then there exists a set F c i - $ d ( X ) such that A c F # X, and, therefore, B - X \ F c i - 8 0 ( X ) \ { ~ } . Clearly, B N A ~.
40
I. Different Families of Sets in Bitopological Spaces
By Definition 1.3.11 for j - i, there is a set V E 7-~\ {~ } such that V c B c 7-~cl V and V n A - ~, which contradicts ~-i cl A - X. [-] The condition (1) of Theorem 1.3.24 and Theorem 1.11 in [68] allow us to define the category notions from Chapter I, the Baire and Baire-like properties from Chapter IV also by means of (i, j)-semi-interiors and (i, j)-semi-closures for BS's and, consequently, to define the category notions and Baire property for TS's too. Proposition (XtT1 < T2) "
1.3.25.
The following statements are satisfied for a BS
(1) If A E (1, 2 ) - $ O ( X ) , UNB-~. (2) If A E (2, 1 ) - S O ( X ) , andUnB-~. (3) If A - U U B, where A c (1, 2 ) - $ O ( X ) . (4) If A - U u B, where A E (2, 1 ) - S O ( X ) .
then A - U U B, where U E 7"1, B E 2-J~f~)(X) and then A -
U U B, where U E T2, B E (1, 2 ) - A f t ( X )
A is 2-connected, U ~ T1 \ {2;} and Bd2 -- ~, then A is 1-connected, U c ~-2 \ {~} and Bdl -- ~, then
Proof. (1) A E (1, 2)-8(9(X) implies that U c A c ~-2 cl V for some V E T1. Hence A - V U (A \ U) and A \ U c T2 cl V \ U c co T2 N 2-Bd(X) c 2 - A f t ( X ) so that A - U U B, where B - A \ U. The proof of (2) is straightforward, proceed as in (1). (3) Clearly, it suffices to show that B c w2 cl U. Contrary: B n (X \ T2 cl U) D#~. ThenB-DUE, whereE-BNT2clU. Hence A-(UUE)
UD,
UUE#~
and U U E c T 2 c l U
so that w2clUNA-UUE
EcoT~
in (A, T[, T~). Since D d C B d, that is, D d - o, it is likewise clear that D E co ~-~. Thus U U E and D constitute the 2-separation for A, which is impossible. Assertion (4) can be proved by analogy with (3). E] Of the above statements on subsets of a BS (X, T1 < 7"2) the first two give the necessary conditions and the other t w o - the sufficient conditions for the belonging of subsets to the families (i, j ) - S O ( X ) . We close this section by introducing new families of subsets of a BS (X, T1,T2) whose elements belong to the meet ( i , j ) - S O ( X ) N (j, i)- S C ( X ) . It is important to note the fact that their corresponding topological family is also new, although the elements of the latter family were casually mentioned as semiclopen (i.e., both semiopen and semiclosed) sets [226]. D e f i n i t i o n 1.3.26. A subset A of a BS (X, T1, T2) is called an (i, j)-semiopen domain if there exists an (i, j)-open domain U c X such that U C A c Tj cl U.
1.3. (i, j ) - B o u n d a r i e s .
(i, j ) - O p e n D o m a i n s a n d . . .
41
The complement in X to an (i,j)-semiopen domain is an (i,j)-semiclosed domain, that is, a subset B of X is an (i, j)-semiclosed domain if there exists an (i, j)-closed domain F c X such that Tj int F C B C F. Hence a subset A of a TS (X, T) is a semiopen domain (semiclosed domain) if there exists an open domain U c X (closed domain F c X) such that U C A c clU ( i n t F C A c F). The families of all (i,j)-semiopen domains ((i,j)-semiclosed domains) of X are denoted by (i,j)-$O:D(X) ((i,j)-$CZ)(X)). Theorem
1.3.27. The following conditions are satisfied for a BS (X,
7"1,7"2)
"
-
and thus for a TS (X, T), we have
s o D ( x ) - sc (x). (2) If A c ( j , i ) - S d ( X ) (A c ( i , j ) - $ O ( X ) ) and
A c TjclTiintTjclA
(TiintTjclTiintA C A),
then A e (i, j)-SOlP(X) - ( j , i)-$ClP(X) and thus for a TS (X, T), we have if A c $C(X) (A E S O ( X ) ) and A c c l i n t c l A ( i n t c l i n t A c A), then A c SOTs(X)Proof. (1) Let A c (i,j)-SOlP(X). that U c A c T j c l U . But
$CZ)(X).
Then there is a set U c (i,j)-OlP(X) such
U E (i, j ) - O D ( X ) ,z---->,U - ~-i int Tj cl U and by Remark 1.3.4, Tj cl U C (j, i)-CD(X),
that is ~-j cl U - Tj cl T~ int ~-j cl U.
Thus for a set A there is a set Tj cl U -- F E (j, i ) - r i D ( X ) such that T, int F C A C F and, therefore, we obtain A c (j, i)-$CD(X). In a similar manner it can be shown that
(j, i)-SCD(X) c (i, j)-SOIP(X) so that (i, j ) - S O D ( X ) - (j, i)-SCD(X). (2) Suppose A c (j,i)- $ C ( Z ) and A c TjclTiintTyclA. By virtue of Theorem 1.3.12, 7~intTjclA c A as A c (j, i)- SC(X). Assume that U TiintTjclA. Following Remark 1.3.4, V c ( i , j ) - O D ( X ) and V c A. Thus U C A c 7j cl U and, taking into account Definition 1.3.26 and (1) above, we find that A e ( i , j ) - S O Z ) ( X ) = (j, i)-8CT?(X). The rest is proved similarly. F1 C o r o l l a r y 1.3.28. Let (X, ~-1,~-2) be a BS. Then
(i, j ) - O D ( X ) U (j, i ) - C V ( X ) C (i, j ) - S O D ( X ) = (j, i)-scz
(x) - (i, j ) - s o ( x )
n (j, i ) - s c ( x )
+
9
"++
m+
II
.
~-~
9
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,..,_.+
+If +-3 .
9
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t-r"J
.+
~
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44
I. Different Families of Sets in Bitopological Spaces
and so Tjint(X\U) cX\AcX\U. Hence X \ A 9 (i, j)-SCD(X)/co AThe proof of the inverse inclusion is similar since the equality co(co A) = A gives that if co A = B, then co/3 = A. The rest of the proof is an immediate consequence of (2) of Corollary 1.3.7 and (2) of Corollary 1.3.15. [] Proposition 1.3.34 allows us to compare the families i-SOD(X) = i-SCD(X) and (i, j ) - S O D ( X ) = (j, i)-$dD(X) by the set-theoretic operation inclusion for a BS (X, T1 <72). 1.4. Sets Pairwise D e n s e in T h e m s e l v e s . ( i , j ) - P e r f e c t Sets and Pairwise Scattered Sets
This section is devoted to the study of other new families of subsets of BS's. Definition 1.4.1. itself if A~ c A].
A subset A of a BS (X, T1,T2) is termed (i,j)-dense in
The families of all subsets of X, (i,j)-dense in themselves, are denoted by (i, j ) - D I ( X ) and the families of all/-discrete subsets of X are denoted by i-Z(X). P r o p o s i t i o n 1.4.2. Let (X, T1, ~-2) be a BS. Then
(1)
1-D27(X) U 2-D27(X) c (1, 2)-DYT(X) = (2, 1)-D27(X)
and for a BS (X, 71 < Ts), we also have (2) 2-D2"(X) c 1 - D I ( X ) = ( 1 , 2 ) - D I ( X ) = ( 2 , 1)-D2:(X), 1-Z(X)c2-2:(X). Proof. It is clear that the inclusion in (1) follows from the equality in (1) and thus, it suffices to prove only the equality. Let A 9 (i, j ) - D I ( X ) . Then A~ c A j so that X \ A j c X \ A~. Hence d i d A n (X \ Aj) c A n (X \ A~), that is A \ Ajd c A \ A~i and Aji c A \ A ii c A~.
Thus A 9 (j, i)-DI(X). (2) The inclusion 71 C T2 implies that A d c A d and A] c A~ for every subset A c X, that is, 2-DI(X) c 1-DZ(X) and 1-I(X) c 2-I(X). By Definition 1.4.1,
A 9 (1,2)-D:r(Z)~
dil c g d.
Hence d~ c d l d implies A~ - ~. Thus A c d l d, that is, A 9 1-DI(X). From this proposition we conclude that if for a BS (X, has at least one l-isolated point, then
T1 <
[~
T2), & set A C X
A ~ 1-D2:(X) = (1,2)-DI(X) = (2, 1)-D2:(X). By the first equality in (1) of Proposition 1.4.2, it suffices to consider only the family p-D:T(X), where p - D I ( X ) = (1, 2)-D2:(X) = (2, 1)-D2:(X).
1.4. Sets Pairwise Dense In Themselves. (i, j)-Perfect Sets and . . .
45
E x a m p l e 1.4.3. Suppose we are given the BS (R, 7-1,7-2), where 7-1 = w is the 1 1 1 natural topology on R, 7-2 is the discrete topology on R and A - {1, 2, 3, 4 "'" } c R. Then AId - {0} and, therefore, A~ - A \ A1d - A, A~ - A and A d - 2~. Hence A gp-TPZ(R). The reverse of the first inclusion in (2) of Proposition 1.4.2 does not hold in general. E x a m p l e 1.4.4. Suppose 7-1 is the antidiscrete topology on R, 7-2 = w is the natural topology on R and a subset A c R is the same as in Example 1.4.3. Then A~ - 2~, A1d - R, A~ - A, and d d - {0}. Hence A~ - ~ implies that A c 1 - ~ P Z ( R ) - p-~PZ(R), but A ~ 2-9 since A~ r ~. P r o p o s i t i o n 1.4.5. For a BS (X, 7-1,7"2) any union of sets p-dense in themselves is p-dense in itself.
Proof. Let {As }s~s be any family of sets p-dense in themselves so t h a t (As)~ (As \ (As)ld) c (As) d for every s E S. It is well known t h a t U (As) d c ( U As) d sCS
sCS
and hence
\( U
U
sES
U
sCS
sES
\ sU CS
U
sCS
(U
s6S
\
2"
l--i
sES
If there exists a set A E i-TP(X) N i-Bd(X), then by virtue of Theorem 4 from [161, p. 83], X c i-T~I(X) C p-T)Z(X). Based on this fact, our next proposition gives examples of new members of the family p-7?:r(X). Proposition 7-1
1.4.6. For every subset A of a BS (X, 7-1 < 7-2), We have
int (1, 2)- Fr A, A n 7-1 int (1, 2)- Fr A c p -T):Z-(X).
Pro@ As is well-known, if U E 7-1 and A c X is any set, then 7-1 cl (U N 7-1 cl A) - 7-1 c l ( g n A). Therefore 7-1 c I ( U n A ) -
7-1 cl (U N 7-2 el A).
Let U -- T1 int(1, 2)-Fr A - 7-1 int(2, 1 ) - F r ( X \ A). Then it obviously follows t h a t T1 cl U - T1 cl(U n T1 cl A) - T1 cl(U N A) and T, C1V -
T 1 C1 ( V n T 2 c l ( X
\ A)) -
T1 c l ( U n ( X \ A)) -
that is Wl cl T1 int(1, 2)- Fr A -
T1 cl(U \ A),
46
I. Different Families of Sets in Bitopological Spaces
=71cl(AnT1int(1,2)-FrA) -71cl(7.1int(1,2)-FrA\A).
(.)
The first equality in (.) shows that the set A N 7.1 int(1, 2)-Fr A is 1-dense in 2)- Fr A. Moreover, (.) also implies that
7"1 int(1,
7.1 int(1, C 7.1
2)- Fr A c 7.1 el (7.1int(1, 2)- Fr A \
el (7-1 int(1, 2)- Fr A \ ( A n
A) c 7.1 int(1, 2)- Fr n))
so that A n 7.1 int(1, 2)-Fr A is the 1-boundary set in 7.1 int(1, 2) Fr A [161, p. 76]. Hence by Proposition 1.4.2 and by virtue of the remark preceding this proposition, we obtain 7.1 int(1, 2)- Fr A c 1-DZ(X) = p-DZ(X). Since A n 7.1 int(1, 2)- Fr A is 1-dense in 7.1 int(1, 2)-Fr A, by virtue of Theorem 3 from [161, p. 83], we obtain A n 7.1 int(1, 2)- Fr A c 1-DZ(X) = p-DZ(X).
D
D e f i n i t i o n 1.4.7. A subset A of a BS (X, 7.1,7.2) is said to be (i,j)-perfect if
Af
A
The families of all (i, j)-perfect subsets of X are denoted by (i, j)-P(X). Hence
(i, j)-~(x)
=
co ~
p-Z~z(x).
n
P r o p o s i t i o n 1.4.8. For a BS (X, 7.1,7.2), we also have i-7)(X) c (i,j)-7)(X), and for a BS (X, 7.1 < 7.2), we have (1, 2 ) - 7 ) ( X ) = 1 - P ( X ) c (2, 1)-7~(X) D 2 - P ( X ) C 1-DZ(X).
Proof. By (1) of Proposition 1.4.2, i-7)(X)-covi n i-DZ(X) C COTi N p - D Z ( X ) - (i,j)-7)(X). Furthermore, if T1 C T2, then by (2) of Proposition 1.4.2, (1, 2)-7~(X) -- co7.1 n p - ~ ) J Z ( X ) - COT1 n 1-DZ(X) -= 1-7~(X) C co 7.2 N p - D : Y ( X ) - (2, 1)-7)(X). Finally, 2 - 7 ) ( X ) - co7.2 N 2 - D Z ( X ) C 2-DZ(X) C 1-D:Z(X) and 2-~(x)
-
c o ~-~ n 2-Z~:Z(X) c c o r
o
p-Z~Z(X) - (2, 1 ) - V ( X ) .
a
The inverse inclusions in the latter proposition are not, generally speaking, correct. E x a m p l e 1.4.9.
Let (]I~, T1, "/-2) be the BS from the previous example. If then A~ - 2~, A d - {0} and hence A~ c A d c A so that A E (2, 1)-7)(R). But A ~ 2 - P ( R ) as A / : A d. It is likewise easy to observe that for the set Z, we have Z~I - 2~, Z d - R, Z~ - Z and Z d - 2~. Hence Z c (2, 1)-7)(R), but Z ~(1, 2)-7)(R) since R - Z1d is not contained in Z. A -
{1, 2,1 3,1 4 , " 1 . , 0 } ,
P r o p o s i t i o n 1.4.10. For a j-T1 BS (X, 71,72), we have j - C I ( p - D Z ( X ) ) c (j,~)-~(x).
Proof. We shall show that A~ c Ajd implies (Tj clA)~ C (Tj clA) ] c zj clA
1.4. S e t s P a i r w i s e D e n s e I n T h e m s e l v e s .
(i, j ) - P e r f e c t
Sets and ...
47
Clearly,
(rj cl A)jd c rj cl(Tj cl A) - rj cl A. Further,
(rj cl m){ - rj cl d \ (rj cl A) d - (d U A d) \ (A U gd) d = = (A U A d) \ (A d U (Ad) d) - ((A O Aj) \ A d) n ((A U __ ((A \ Ad) u (Ad \ Ad)) N ((A \ (Aj)i) Ud
AJ) \/AJ)f)
(Ad \ (Aj)~))dd .
Since A d \ A d c Aad and A \ A d - A~ C A d, we conclude that the union in the first brackets above is contained in A d. Hence the set (rj cl A){, being the intersection, is completely contained in Aad. But A d - ( r j cl A) d and, therefore, (rj clA)~ c (rj clA) d c rj clA.
[-1
Using the equalities (1, 2)-T)27(X)= (2, 1 ) - D Z ( X ) = p - D Z ( X ) , we introduce D e f i n i t i o n 1.4.11. A subset A of a BS (X, rl, r2) is termed p-scattered if A is nonempty and contains no nonempty p-dense in itself subset. The families of all p-scattered subsets of X are denoted by p - S T ( X ) . The latter definition and Proposition 1.4.2 readily yield P r o p o s i t i o n 1.4.12. The following conditions are satisfied for a BS (X, rl, r2):
(1) p - s ~ - ( x ) c ~ - s r ( x ) n 2-S~-(X). (2) A c p - S T ( X ) and B c A, B ~ ~, imply B c p - S T ( X ) . For a BS (X, rl < r2), we also have (3) 1-ST(X) = p-ST-(X) C 2-87"(X). R e m a r k 1.4.13. Let (IR, rl, r2) be the BS from Example 1.4.4, where
A-
1, 2' 3 ' 4 "
E 1-DZ(N) -p-DZ(N).
If ~ r B c A is any subset, then B c
1-Z~Z(R) = p-Z~Z(R) and so _4 g 1-8~r(R) = p - S ~ r ( R ) .
But B c A implies B~ C A~ - {0} and, therefore, B N B ~ -- 2~. B ~ 2-DZ(R), and since B C A is arbitrary, we obtain A c 2-ST(R).
Hence
Also note that following Example 1.4.4 and the above remark, we have
A-
{111
1,2, a,4, - -
} ~
~
~
c 2-I(R)
~
Thus, generally speaking, the family i-Z(X) is not contained in p - S T ( R ) since A g p - S T ( R ) . However, we have P r o p o s i t i o n 1.4.14. If for a BS (X, rl, r2) a set A c i-Z(X) and for every subset B c A we have B-g j-Di[(X), then A ~ p - S T ( X ) .
Proof. Assume that A c i - l ( X ) and B C A is an arbitrary subset. Then B c i-Z(X), and B - g j - D Z ( X ) implies that Bji # ~. Since B N B d - ~ and Bji c B, we
48
I. Different Families of Sets in Bitopological Spaces
obtain Bji n B d - Z so that B - ~ p - D Z ( X ) . Therefore A E p - S T ( X ) is arbitrary.
since B c A D
P r o p o s i t i o n 1.4.15. Let (X, 7-1,7-2) be a Rep-T1 BS. Then X = A U ( X \ A), where A c (1, 2 ) - 7 ) ( X ) n (2, 1)-7)(X) and X \ A c p - S T ( X ) (it stands to reason that one of these subsets may turn out empty). Proof. Suppose A = U{E : E E p - D Z ( X ) } . Then by Proposition 1.4.5, A c p - D Z ( X ) . It follows from Definition 1.4.7 and Proposition 1.4.10 that 7-j clA E (j, i ) - P ( X ) c p - D r ( X ) . Recalling that A is maximal, we obtain A c co 7-1 n CO 7"2 and, therefore, A
=
7-1
clA = 7-2 clA E (1, 2 ) - 7 ) ( X ) n (2, 1)-7)(X).
It is likewise clear that X \ A contains no nonempty p-dense in itself subset and thus X \ A E p - S T ( X ) . D The latter proposition shows that the maximal p-dense in itself subset of a BS (X, 7-1,7-2) is both 1- and 2-closed. P r o p o s i t i o n 1.4.16. For a BS (X, have (1, 2)-Fr A E (1, 2)-AfD(X).
7-1 < 7-2)
and every set A c p - S T ( X ) ,
we
Proof. By Proposition 1.4.6, the set ANT-1 int(1, 2)- F r A c p - D Z ( X ) and, hence, is empty in view of the fact that A E p - S T ( X ) . Moreover since ANTI int(1, 2)- Fr is 1-dense in 7-1int(1, 2)- Fr A, the latter set is also empty, that is, ( 1 , 2 ) - F r A co 7-2 N 1-Bd(X) C (1, 2)-AfD(X).
it A c D
Note that due to Propositions 1.4.2 and 1.4.12 for a BS (X, 7 1 ( 7 - 2 ) all results that hold for the families 1-DZ(X) and 1 - S T ( X ) also hold for the families p - D r ( X ) and p - S T ( X ) , respectively. We finish the discussion of (i,j)-perfect sets by proving a bitopological modification of Cantor-Bendixson's theorem. In this context recall that for a BS ( X , 7-1 < 7-2) a set A c p - D Z ( X ) if and only if U c 7-1 and U N A =fi 2~ implies that U n A is infinite, and a point x E X is an/-condensation point of a set A c X if e a c h / - o p e n neighborhood U(x) meets A in an uncountable set. The set of all /-condensation points of A is denoted by A ~ T h e o r e m 1.4.17. Let (X, 7.1 < 7.2) be a l-T1 and 1-second countable BS. Then any uncountable 2-closed set F contains a set A E co7.2 n p - D r ( X ) = (2, 1)-P(X). Moreover, F ~ = F ~ = T1 clA. Proof. Let F E cot2 be an uncountable set and U1, U2,... be a countable base of 1-open sets. Suppose that V1, V2,... be those of the sets U1, U2,... that meet F in a countable set. Clearly, the sequence V1, V2,... may be finite or infinite. If (X)
A-
F \ U vk, then rl c r2 implies A c co r2 and k=l OO
OO
u
(u
k=l
k=l
(N:)
(U k=l
1.4. Sets P a i r w i s e D e n s e In T h e m s e l v e s .
(i, j ) - P e r f e c t Sets a n d . . .
49
oo
oo
Since F is uncountable and U vk a F is countable, we obtain t h a t A - F \ U vk k=l
k=l
is uncountable and, therefore, non-empty. It remains only to prove t h a t A c p-7?Z(X). But we would rather prove t h a t A c A ~ that is, each point z c A is a 1-condensation point of A so t h a t U(z) E T1 gives I N ( z ) N A[ > R0. Indeed, let U(x) c T1 be any neighborhood. Then there is a set Un from the 1-countable base 0"1, 0"2,... such t h a t z c Un c U(x). It is obvious t h a t Un /: Vk for each k since the contrary implies z g F \ U vk = A. k=l
This means that IF a U~] > R0. Since A differs from F merely in a countable set, it follows that [ANU~ I > R0. But A N U ~ c A N N ( x ) so that I A N U ( x ) [ > b~0 and so A c A ~ Therefore A c AId and thus A c co ~-2 n p-7927(X) - (2, 1)-7)(X). Now, we have A c A ~ c A d so t h a t T1 C1 A C T1 C1 A~ -
Therefore A ~
-
7-1
A 0 C T1
cl A d - A d c 71 cl A .
cl A. Finally, (x)
(3o
-1~
1
k=l
k=l
1
C o r o l l a r y 1.4.18. Under the hypotheses of Theorem 1.4.17, for any uncountable set B C X there is an uncountable subset A C B such that A c p-T)Z(X).
Proof. Let B c X be any uncountable set. Then the result follows directly from the proof of Theorem 1.4.17 omitting the remark that A is 2-closed. D At the end of this section we shall consider three operators on 2 x , which are used to characterize degrees of nearness of the four boundaries of a set, the S-, C- and N-relations in Chapter II and interrelations of dimension functions in Chapter III. D e f i n i t i o n 1.4.19. For a BS (X, T1 < ~-2) the indicators of nearness of the boundaries are the following three operators: n l , n2, n : 2 X --~ (2, 1)-12C(X), defined as follows: nl
(A) = T1 cl A \ 7-2 cl A, n2(A) = ~-2 int A \ T1 int A
and n(A) - n l ( A ) u n2(A) for each set A c 2 X. It is obvious t h a t n~(A) - n j ( X \ A) so t h a t n(A) - n ( X \ A) for each set A c 2 X, the restrictions nllCOWl
-- n2171
--
n
71neoY 1 z
and, therefore, nlw I - n l ,
n[co~-i - n 2 .
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1.5. Relative Properties
51
If A 9 (72 \ 7"1)n CO 7-1, then n ( A ) -- n 2 ( A ) - A N T1 c l ( X \ A) - T1 c1A AT1 c l ( X \ A) - 1 - F r A -
=T2clANTlcl(X\A)-(2,1)-F'rA, If A
r (~-2 \ 71) N (co 7-2 \ co 7-1),
(1, 2)- Fr A - 2- F r A - 2~.
then
n l ( A ) - 71 c I A n ( X \ A) - 7-1 c l A n
T2 c l ( X \ A) - (1, 2)- E r A ,
n 2 ( A ) - A n T1 c l ( X \ A) - T2 C1A n T1 c l ( X \ A) - (2, 1)- Fr A,
n(A)-(1,2)-FrA
U (2,1)-FrA-(1,2)-FrAA(2,1)-FrA-I-FrA,
2-FrA-2~.
(3) The proof consists of elementary calculations taking into account the fact that the equality nl(A) - n2(X \ A) is fulfilled for each set A 9 2 x. (4) If A 9 p-TPZ(X), then A~ c A d - (A1d \ A d) U A f so that A; c (A d \ A d) - (A U A1d) \ (A-U A d) - nl (A). Conversely, A~ c nl(A) implies A~ c A1d \ A~ c A1d and so A 9 p-7:)Z(X).
[]
1.5. R e l a t i v e P r o p e r t i e s Relative properties, that is, to say, such properties of subsets of subspaces of TS's that are preserved from spaces to subspaces, from subspaces to spaces or in both directions, were investigated in various published works on general topology. Naturally, there arises a question how widespread relative properties are in the theory of BS's because these properties will play an important role in our further investigations. D e f i n i t i o n 1.5.1. Let (X, T1,7-2) be a BS and (Y, r~, r~) be a BsS of X. If A c Y, then (1) A c (i, j)-D(Y) if one of the following equivalent conditions is satisfied:
Y-T~clTjclANY
~
Y c r~ clw] clA ~
Y - r~'clw] clA.
(2) A c (i, j)-Bd(Y) if one of the following equivalent conditions is satisfied: Y - 7-~cl(Y \ ~-~int A) N Y <--> Y C 7-i cl(Y \ 7-] int A) < :, < :- Y - ~-: cl(Y \ T] int A) ~
~-~int r~ int A - 2~.
(3) A c (i, j)-A/'TP(Y) if r] clA c i-13d(Y)so that if one of the following equivalent conditions is satisfied: Y - Ti cl(Y \ ~-] el A) N Y +---> Y C ~-i cl(Y \ ~-] el A) < > Y - r~ cl(Y \ 7-~ cl A) <--->, T~ int r] cl A - 2~. (4) A c (i, j)-SlP(Y) if one of the following equivalent conditions is satisfied: Y r ~-i cl(Y \ Tj cl A) n Y <-->, Y \ T~cl(Y \ T] cl A) / 2~ ,' < ;-Y # T/cl(Y \ T ] c l A ) ~
T~intT~clA =/= 2~.
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53
C o r o l l a r y 1.5.4. Let (X, 7.1 ( 7 " 2 ) be a BS, (7.2clY, 7.~,7.~)and (Y, 7.~',7.~') be BsS's of X , and A c 7.2 cl Y. Then the following conditions are satisfied: A c (2, 1)4VD(~2 d Y) --*, A n Y c (2, 1)-ArD(Y) so that
A n Y c (2, 1)-$/P(Y) ---> A c (2, 1)-S:D(% cl Y) and thus A E (2,1)-Catgi(7.2clY) ~
A N Y c (2,1)-Catgi(Y),
A N Y E (2,1)-Catgii(Y)
A c (2,1)-Catgii(7.2clY).
~
Proof. It suffices to show only the validity of the first implication. By Proposition 1.5.3, Ac(2,1)-AfZ~(7.2clY),z--->7.; c l A c % el (%elY \ 7.~ c l A ) - % el ( % e l y \
7.1
clA),
while (2) of Lemma 0.2.1 gives 7.; cl A c 7.2 cl (7.2 cl Y \ % el 7.1 el A) C 7.2 cl(Y \
7.1
cl A).
Hence 7.; cl(A n Y) c % cl(Y \ 7.; cl A) c % cl (Y \ 7.; cl(A n Y)). But A n Y c Y c T2 cl Y ~
T 1" c l ( A n Y ) - -
T 1' c l ( A N Y )
NY
and thus T;' cl(A N Y) C 7.2 cl (Y \ T; cl(A N Y)) - T2 cl (Y \ T;' cl(A N Y)). Hence, again applying Proposition 1.5.3, we find that A n Y c (2, 1)-A/T)(Y).
D
By analogy with the second part of Theorem 1.1.3, we obtain the sufficient conditions for the relative (i, j)-nowhere densities. P r o p o s i t i o n 1.5.5. Let (X, 7-1,72) be a BS and (Y, 7.~, 7.~) be a BsS of X, where Y c j-7?(X) and A c Y . If for every set V c 7.~ \ {~} there exists a set V e vj \ {~} such that V n A = 2J, then A e (i,j)-A/Z)(Y). For every set U' c T~\{2~} there exists a set U E Ti\{2~} such that U N Y = U'. Hence, by assumption, there is a set V ~ rj \ {2~} such that V c U and V N A = ~. But r j c l Y = X implies that V N Y = V' C r j \ { ~ } and so it remains for us to use the second part of Theorem 1.1.3. D Pro@
7-1,7-2) be a BS and (Y, 7.~, 7.~) be a BsS of X , where Then the following statements hold:
T h e o r e m 1.5.6. Let (X, Y c 7.i and A c X .
A ~ ( i , j ) - A f Z ) ( X ) --->, A n Y E (i,j)-Af:D(Y) so that A n Y c ( i , j ) - S Z ) ( Y ) ----> A E ( i , j ) - S T P ( X ) and thus A c ( i , j ) - d a t g i ( X ) --->, A N Y C (i,j)-Catg~(Y), A n Y C (i,j)-Catg~i(Y) ~
A c (i,j)-Catgii(X).
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C H A P T E R II
D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s on a Set a n d B i t o p o l o g i c a l I n s e r t i o n s In Sections 2.1-2.3 we consider the relations on a set, reducing emphasis on points and focusing attention on families of sets, namely on topologies. At first glance one might think that our reasoning goes beyond the framework of this monograph, but in the concluding section we recapitulate the general principles so as to characterize the C- and N-relations through bitopological insertions. The S-relation was introduced by A . . R Todd as an equivalence relation and it expresses a close relationship between two topologies on a set [252]. The coupling of topologies, that is, the C-relation was defined by J. D. Weston to generalize some well-known theorems on topological groups and linear TS's and to connect the same properties of the coupled topologies [258]. The so-called nearness of topologies, that is, the N-relation is defined for the first time in [95]. Along with the above-mentioned relations and their equivalent characterizations, we also consider their combinations with the set-theoretic operation inclusion. In other words, we consider the relations < s , < c , and < x . In contrast to the S-relation, the coupling and nearness of topologies are not symmetric relations, though, topologies on the same set are partially ordered by the relations < c and
The interdependence of the above three relations and the relative properties as well as the relationship between various families of subsets of a given BS are investigated in the case of S-related, <s-related, coupled,
63
64
II. Different Relations Between Two Topologies . . .
2.1. T h e S - R e l a t i o n D e f i n i t i o n 2.1.1. A family B of subsets of a TS (X, 7) is a pseudobase of the topology ~- if the following conditions are satisfied: (1) B c B ----5, int B ~= 2~. (2) For every subset U c 7- \ {~}, there exists a set B E B such t h a t B c U.
By [252] this definition leads to the equivalence relation on the family of all topologies on a set X. D e f i n i t i o n 2.1.2. Topologies 71 and ~-2 on a set X are S-related (briefly, TIST2) if T1 \ {~} is a pseudobase for 72. A set X together with the S-related topologies T1 and ~-2 is denoted by ( X , T1 S T 2 ) .
R e m a r k 2.1.3. The S-relation between two topologies on a set is especially i m p o r t a n t by Proposition 3.4 from [252], following which if one member of an S-equivalence class is Baire, then all members of this class are also Baire. E x a m p l e 2.1.4. Let ( R , s , v ) be a BS, where s is the half-open interval topology, that is, the Sorgenfrey topology on R and so the basic open sets for s are of the form [a, b), and r is the topology with basic open sets of the form (a, b]. It is clear that neither topology is finer than the other, the greatest lower bound inf(s, r ) = s n r = ~ of the topologies s and r is the natural topology on R, and the least upper bound sup(s, T) of the same topologies is the discrete topology on R. Moreover, s S r , s S i n f ( s , r ) and v S i n f ( s , r ) . Hence, by Remark 2.1.3, the Sorgenfrey line, that is, R with the Sorgenfrey topology is a Baire space since s S i n f ( s , r ) and the natural topology is completely metrizable and, therefore, Baire [252]. Theorem
2.1.5. The conditions below are equivalent for a B S (X, T1, T2):
(1) T1 is S-related to T2.
(2)
1- d(X) = 2 - B d ( X ) = (1,
(2,
so that
= 2-z,(x)=
(8)
(1, 2 ) - z , ( x ) =
T l i n t A C TlClT2 int A A T2 int A C T 2 c l T I I n t A
SO that T1
int ~-2 cl A c T1 cl A A w2 int
T1
cl A c 7-2 cl A
for every subset A c X . (4)
71 C (2, 1 ) - S O ( X ) A 7-2 C (1, 2 ) - $ 0 ( X ) ,
so that COWl C (2, 1)-,SC(X)A cow2 C (1, 2)-,S'C(X).
2.1. The S-Relation
65
Proof. It is clear that =
-' > (7-1int A ~ 2~ ~
,,.
;.
7-2 int A ~= 2~ for every subset A c X ) .
Thus, by Proposition 1.1.11, for the equivalence (1) ,z--> (2) it is enough to show that (1) ,z--->, (7-1 int A r ~ ~ 7-2 int A ~ ~ for every subset A c X). We assume that 7-1S7-2, that is, 7-1 \ {2~} is a pseudobase for 7-2, and A c X is any subset. If 7-1 int A ~= z , then by (1) of Definition 2.1.1, 7-2 int 7-1 int A # ~, and hence 7-2 int A # ~. When 7-2 int A # z , by (2) of Definition 2.1.1, there exists a set V E 7-1 \ {Z } such that V c 7-2 int A. Thus 7-1 int 7-2 int A # 2~ so that 7-1 int A # ~. On the other hand, let U e 7-1 \ {2~}. Then 7-2 int U :/= 2~. If V e 7-2 \ {2~}, we have ~ r V = 7-1 int U c U and, consequently, 7-1 \ {2~} is a pseudobase for 7-2. Therefore (1) .z--> (2). The implication (1) ~ (3) is exactly Proposition a.a from [252], (2) is obvious. (3) ,z----5, (4) is an immediate consequence of Theorem 1.3.12. D It is likewise easy to see that if 7-1 = a~ is the natural topology on R and 7-2 is the discrete topology on R, then 7-1 is not S-related to 7-2. In the sequel it will be assumed that (7-1S7-2 A 7-1 C 7-2) ~ 7-1 < s 7-2 and the corresponding BS will be denoted by (X, 7-1 < s 7-2). Furthermore, Example 2.1.4 shows that for a BS (X, 7-1 < s 7-2) the equality 7-1 = 7-2 does not hold in general. C o r o l l a r y 2.1.6. The following conditions are satisfied for a BS (X, 7-1S7-2): 9
(1) X[ - X~ and thus if either of the topologies
7-1 and 7-2 is discrete or antidiscrete, then 7-1 = 7-2. (2) (X, 7-1,7-2) is d-quasi regular if and only if (X, rl, 7-2) is p-quasi regular. (3) If (Y, 7-~,7-~) is a BsS of X , where Z 9 (7-1 N 7-2)[-J 1 - ~ ) ( X ) 2-~:)(X), then 7-~S7-~.
Proof. The proof of (1) is straightforward. (2) We begin by assuming that (X, 7-1,7-2) is a 1-quasi regular and 2-quasi regular BS and U c 7-~\ {~}. By t h e / - q u a s i regularity of (X, 7-1,7-2), there is a set V E 7-~\ {~} such that 7-~cl V C U. But 7-1S7-2 implies that 7-j int V = W r ~ and by the j-quasi regularity of (X, 7-1,7-2), there exists a set E c rj \ {~} such that rj c l E C W. Let O = 7-~int E. Then O r ~ and 7-j cl O c U so that (X, rl, 7-2) is (i, j)-quasi regular. Conversely, if (X, 7-1,7-2) is (1,2)-quasi regular and (2, 1)-quasi regular, then for a set U c r~ \ {~}, there is a set V c 7-~ \ {~} such that 7-j el V c U. For the set r j i n t V = W r ~, there is a set E c rj \ {~} such that r ~ c l E c W. Let O = 7-i int E. Then O r ~ and ri cl O c U so that (X, 7-1,7-2) is/-quasi regular. (3) First we assume that Y E 7-1 N 7-2 and U E 7-~ in (Y, 7-~,7-~). Then U E 7i and by (4) of Theorem 2.1.5, U c (j, i ) - $ O ( X ) . Hence, according to (1) of Proposition 1.5.26, U c (j, i ) - S O ( Y ) so that 7-~ c (j, i ) - S O ( Y ) and, again applying (4) of Theorem 2.1.5, we obtain 7-~$7-~. On the other hand, let Y c 1-Z)(X) = 2-79(X) and A c Y, r~ int A/= ~. Then, there is a set U c 7-i such that U ~ Y = 7-~int A. But 7-1S7"2 implies 7-j int U = V r 2~
66
II. Different Relations Between Two Topologies . . .
and, therefore, V Cq Y = V' # ;~ as Y 9 1-D(X) = 2-D(X). Thus Tj int A # since V' C r" int A. D C o r o l l a r y 2.1.7. The conditions below are equivalent for a BS (X, rl, r2): (1) rl < s v2. (2) 1-13d(X) c 2-13d(X) so that 1-D(X) C 2-D(X). (3) r2 int r2 cl A = re int T1 cl A so that r2 cl r2 int A = r9 cl r l i n t A
for every subset A c X . (4) r2 c (1,2)-SO(X) so that cot2 c (1,2)-SC(X). (5) n(A) 9 2-Bd(X) for each set A 9 2x.
Furthermore, the next statement holds: (6) If (Y, 7-;, r~) is a BsS of X and Y 9 r2, then T 1 S T 2 -----5, T 1
Proof. The equivalences (1) <---> (2) ,e--->, (3) <--> (4) are immediate consequences of the corresponding equivalences of Theorem 2.1.5. (1) ------5, (5). If 7"1S7"2, then by (b) of 4.A.2 in [173], nl(A) 9 2-A/'D(X) and n2(A) 9 2-A/'D(X) for each set A c 2 x. Therefore n l ( A ) c 2-Bd(X) and n2(A) E 2-Bd(X) for each set A E 2x, and thus n(A) c 2-Bd(X) for each set A E 2x. (5) ---> (4). Clearly, if n(A) 9 2-Bd(X) for each set A 9 2x, then n2(A) 9 2-Bd(X) for each set A 9 2 x. Hence ~-2 int(r2 int A \ r l i n t A) = ;~ for each set A c 2 x so that r2 int A c r2 cl r l i n t A for each set A c 2X, that is ~-2 C (1, 2)-$(.9(X). (6) Let A c Y and r ~ i n t A ~= ~. Then r 2 i n t A # ~ and, by (2) above, r l i n t A ~: 2~. But r l i n t A c r~ int A and so r; Sr~. [5] Further, following (1) of Corollary 1.3.15 and (4) of Theorem 2.1.5, for a BS have
( X , T1, T2), w e
rl < s r2 ~
(rl c r2 c (1,2)-,,gO(X)c ( 2 , 1 ) - $ 0 ( X ) ) .
Note that in contrast to (3) of Corollary 2.1.6 and (6) of Corollary 2.1.7 the S-relation is not, generally speaking, hereditary with respect to/-closed subsets. E x a m p l e 2.1.8. Let X = {a,b,c,d}, 71 = {Z, {a, b}, {a, b, c}, X } and 72 = { z , { a , b } , { a , b , c } , { a , b , d } , X } . Clearly, rl < s r2, but if F = {c,d} e cot1 c co 72, then r~ is not S-related to v~ for the BS (F, v~, v~). This fact leads to the following notion to be used in Chapter IV.
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68
II. D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s . . .
Proof. To establish (1)-(4) observe t h a t it suffices to prove only the corresponding first conditions. (1) If A c i-N'D(X) and A-~(i,j)-AfD(X), then by (3) of T h e o r e m 2.1.5, # 7.~ int 7.j cl A c 7.~ el A so t h a t 7.~ int 7"~el A # ~, which is impossible. Conversely, if A E (i, j)-N'D(X) and A-~ i-N'D(X), then (3) of T h e o r e m 2.1.5, used for the set 7"i el A, yields the inclusion 7"i int 7"i cl A c 7"i cl 7"j int 7"i cl A and, therefore, 7"i int 7"i cl 7"j int 7"i cl A # 2~. Hence (2) of T h e o r e m 2.1.5 implies t h a t 7"j int 7"i cl 7"j int 7"i cl A # o and by L e m m a 0.2.1, 7.j int 7.~ cl 7.j int 7.~ cl A = 7.j int 7.i cl A # ~. But 7"j int 7"i cl A = 7"j int 7"j int 7"i cl A # 2~ and by (2) of T h e o r e m 2.1.5, we conclude t h a t 7"~int 7"j int 7"~cl A # ~. Thus 7"~int 7"j cl A # ~, t h a t is, A --~(i, j)-N'D(X) since by (3) of T h e o r e m 2.1.5, 7"j int 7"i clA c 7"j clA. The obtained contradiction shows, firstly, t h a t 7"j int 7"i cl A = 2~, t h a t is, A c (j, i)-N'D(X) and, secondly, t h a t 7"~int 7"~cl A = ~ so t h a t A E i-N'D(X). (2) If A c 2-SO(X), then from T h e o r e m 1.3.12 for i = j = 2 and (3) of T h e o r e m 2.1.5, we infer t h a t A c 7.2 clT.2intA c 7"2 c17.1 i n t A and so A c (1, 2 ) - S O ( X ) . If A E 1-80(X), then from T h e o r e m 1.3.12 for i = j = 1 and (3) of Theorem 2.1.5, we infer t h a t A c T1 cl T1 int A C T1 el 7.2 int A and so A E (2, 1 ) - S O ( X ) . (3) T h e first equality and the inclusion follow immediately from (2) above and (1) of Corollary 1.3.15. If A c (2, 1)-SO(X), then from T h e o r e m 1.3.12, (3) of T h e o r e m 2.1.5, and (2) of L e m m a 0.2.1, we have A c 7"1 cl 7"2 int A c 7"1 cl 7"2 cl 7"1 int A = 7"1 cl 7"1 int A and so A c 1-SO(X). Thus, by (2) above (or, by ( 1 ) o f Corollary 1.3.15) (2, 1 ) - 8 0 ( X ) = (4) Let A c X be any subset. Then by (3) of Corollary 2.1.7, A = 7"2 int 7"2 cl A ~
1-80(X).
A = 7"2 int 7"1 cl A.
Hence, by Definition 1.3.3, 2 - O D ( X ) = (2, 1)-OTP(X). Furthermore, 7"1 el 7"2 el 7"2 int A = 7"1 cl 7"2 cl 7"1 int A and by (2) of L e m m a 0.2.1 T1 cl 7"2 int A = 7"1 c17"1 int A so t h a t A = 7.1 cl T2 int A ,z-->, A = 7.1 cl 7"1 int A and so (1,2)-CD(X) = 1 - C D ( X ) or, equivalently, (1,2)-OD(X) = 1-OD(X).
a
E x a m p l e 2 . 1 . 1 1 . Let (X, T1,7.2) be the BS from E x a m p l e 1.3.8. T h e n T1 <S T2, {b,c} C 1 - S O ( X ) = (2, 1 ) - S O ( X ) , but { b , c } c 2 - $ O ( X ) = (1, 2 ) - $ O ( X ) , {a} ~ 1-OD(X) : ( 1 , 2 ) - O D ( X ) , and { a } ~ 2 - O D ( X ) = (2, 1)-OD(X). Also, {a,b} E (2, 1)-OD(X)= 2-OD(X), but {a,b} -4 (1,2)-OD(X) = 1-OD(X). C o r o l l a r y 2 . 1 . 1 2 . Let (Y, 7.~,7.~) be a BsS of a BS (X, T1 <S T2) and A c Y. If Y E 7-2 U 2 - D ( X ) , then
9
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70
II. D i f f e r e n t R e l a t i o n s B e t w e e n T w o T o p o l o g i e s . . .
Proof. Note that, following (3) of Corollary 2.1.6 and (6) of Corollary 2.1.7, in all cases (1)-(8), the <s-relation is hereditary with respect to the corresponding BsS (Y, ~-~ < w~). Hence the coincidence of the families of subsets of a BsS (Y, ~-~ < s r~) in (1)-(6) are immediate consequences of (1)-(4) in Theorem 2.1.10. For the rest,
we have: (1) According to the well-known topological fact, Y c r2 U 2-D(X) implies that A E 2-A/'D(Y) <---> A c 2-A/'D(X). (2) By (2)of Corollary 1.3.7, T1
N
(2, 1)-OD(X) = (1, 2)-OD(X)N (2, 1)-OD(X).
Hence Corollary 1.5.22 gives A e (i,j)-OD(Y) <---> A e (i,j)-OD(X). Assertions of (3) are immediate consequences of Proposition 1.5.25. (4) The inclusions in this condition follow directly from (3) of Theorem 2.1.10 and the equivalences are evident by Corollary 1.5.27. Assertions of (5) are obvious by Corollary 1.5.ao and (3) of Theorem 2.1.10. The conditions of (6) follow directly from (2) of Corollary 1.a.r and Corollary 1.5.32. Finally, by well-known topological facts, Y c r2 implies the equivalence A c 2-Bd(Y) e---->,A c2-13d(X) and Y c 2-D(X) implies the equivalence A E 2-D(Y) e > A E 2-D(X). Thus (7) and (8) are immediate consequences of (2) of Theorem 2.1.5. It is obvious that by (2) of Theorem 2.1.5, in (1) and (8) we can consider r2U1-D(X) and ruN1-D(X) instead of r2U2-D(X) and T2N2-D(X), respectively. E] C o r o l l a r y 2.1.13. Let (X, rlSr2) be a BS and A E rl U COrl U r2 U cot2. Then
( i , j ) - F r A c 1 - H D ( X ) = 2-HZ)(X)= (1, 2)-AYD(X) = (2, 1)-HD(X).
In addition, for a BS (X, rl < s T2), we also have: (1) If A c X is any subset, then n ( d ) = n l ( A ) U n2(A) e 1 - A f D ( X ) = 2-A/'D(X) = (1, 2)-A/'D(X)= (2, 1)-A/'D(X). (2) X is of 1-datg II <---> X is of 2-datg II e---> X is of (1, 2)-Catg II e---> X is of (2, 1)- CatgII ---> 1- G~(X) N 2-D(X) = 1- Ga(X) N 1-D(X) c 2-Ga(X)N1-D(X) = 2 - G a ( X ) N 2 - D ( X ) C 2-Catgut(X)= 1-Catg~i(X ) = (1,2)-Catgix(X) = (2, 1)-CatgI~(X ).
Proof. First assume that A c X is any subset. Then by (3) of Theorem 2.1.5, we have ~-i int rj cl(i, j)- Fr d = ~-i int rj cl (ri cl A a rj cl(X \ d)) c c T~int rj cl ~-~cl A a r~ int rj cl(X \ A) c ~-~cl A N T~cl(X \ A) = i- Fr A. Similarly, riint rj cl(j, i)- Fr A c i- Fr A. If A E T~ U co r~, then by Theorem 3 from [161, p. 74], i-FrA E i-A/'D(X). Hence by (1) of Theorem 2.1.10 and the inclusions above, riint rj cl(k, 1)- Fr A e 1-HD(X) = 2-A/'D(X) = (1, 2)-HD(X) = (2, 1)-A/'D(X)
2.1. The S-Relation
71
and so 7-i int 7-j cl 7-i int 7-j cl(k, l)- Fr A = 2~, where k, l E {1, 2}, k ~ 1. Thus by (1) of Lemma 0.2.1,
(i,j)-Fr A c 1-AF/?(X) = 2-AF/9(X) = (1, 2)-A/'/9(X) = (2, 1)-AF/9(X) for each set A c 7-1 d co 7-1 U 7-2 U co 7-2. (1) It suffices to show that n l ( A ) E 1-Af~D(X) since n2(A) = n l ( X \ A). Obviously, T 1 int 7-1 cl
(7-1 cl A \
7-2 cl
A) c
and, by (3) of Theorem 2.1.5 for
7-1 int 7-1 cl
T 1 C 7"2,
7-1 int
A c~ (X \
T1
cl 7-1 int 7-2 cl A)
we have
rl cl A C 7-2 cl A.
Hence T1
int
7-1 cl
A c 7-1 cl 7"1 int 7-2 cl A
so that T1 int T1 cl A c~ (X \ q cl 7-1 int r2 cl A) = 2~ and thus n 1 ( A ) --- 7-1 cl
A \ 7-2 cl A ~ 1-A/'Z)(X).
The proof of (2) follows directly from Proposition 1.1.26, (2) of Theorem 2.1.5, and (1) of Theorem 2.1.10. [-] R e m a r k 2.1.14. According to [252] let us treat a TS (Y,'7) as the image of a TS (X, r2) under a continuous function f with the property: for every set U c 7-2 \ {2~}, there exists a set V c 7 \ {2~} such that f - l ( v ) C U. Clearly, the family 7-1 = f - l ( ~ ) = { f - l ( p ) : V E ~ } is a t o p o l o g y o n X c o a r s e r t h a n 7-2. I f w e denote this relation between the topologies 7-1 and 7-2 on X by 7-1f72, then 7-1fr2 is a stronger connection between the topologies than the <s-relation. Hence all results obtained for the <s-relation also hold for the f-relation. Furthermore, following [201], a condensation is a one-to-one and continuous function f : (X, 7-2) ~ (Y, "y) such that I ( X ) = Y. It is clear that in this case f : (X, 7-1) --+ (Y, 3') is a homeomorphism and, conversely, for any BS (X, 7-1 < ~-2) the identity function f : (X, 7-2) ---' (X, 7-1) is a condensation. Theorem
2.1.15. If f : (X, 7-2) ---' (Y, 7) is a condensation and 7-1 = f - l ( . y ) ,
then 7-1 <s 7-2 ,z----5, TlfT-2 ~
f is feebly open
(see Definition 5.1.37). Proof. By Remark 2.1.14 for the first equivalence, it suffices to prove that T1 < s T2 implies TlfT2. Let U c T2 \ {2~} be any set. Then T1 int U c 7-1 \ {2~} and, hence, there is a set V c 7 \ {2~} such that f - l ( V ) = 7-1 int U C U, that is, 7-1f7-2. Now, let 7-1f7-2 and U ~ 7-2 \ {2~} be any set. Then there is a set V E 3' \ {2~} such that f - l ( V ) c U. Therefore f(f-l(v))
= V c f(U),
that is Tint f ( U ) r Z
72
II. Different Relations Between Two Topologies ...
and thus f is feebly open. Conversely, let f be feebly open and U c 7"2 \ {~} be any set. Then the set Tint f(U) - V e T \ {2~} satisfies the inclusion f - l ( v ) C U and so 7"1f7"2. E] C o r o l l a r y 2.1.16. If f " (X, T2) ~ (Y, T) is a feebly open condensation, (Y, T) is quasi regular, 7.1 - f - l ( T ) and (X, 7.1,7.2) is 1-quasi regular, then (X, 7.1,7"2) is d-quasi regular and p-quasi regular.
Proof. It is obvious that 7"1f7"2 and by Proposition 3.5 from [252], (X, T1,7-2) is 2-quasi regular. Therefore (X, 7.1,7"2) is d-quasi regular and by (2) of Corollary 2.1.6, (X, T1,7"2) is p-quasi regular since T1 < S 7"2. [-] By the above reasoning, if for a BS (X, 7"1 < 7"2) the identity function (i.e., the inclusion function) j ' ( X , 7.2) ~ (X, 7 - 1 ) i s feebly open and (X, 7.1,7.2)is 1-quasi regular, then (X, 71, ~-2) is d-quasi regular and p-quasi regular. 2.2. T h e C - R e l a t i o n Let (X, 7.1,7.2) be a BS. Then it is obvious that z2 c 7"1 ~ 7"1 cl A c 7"2cl A for every subset A c X. In [258] a topology 7"1 is chosen among different subfamilies of 2 x whose elements satisfy inclusions of the type given on the right-hand side of the above equivalence. This section continues the study begun in [258], while the Section 2.3 will deal with the same inclusion for a topology 7"2. D e f i n i t i o n 2.2.1. A topology T1 is coupled to a topology 7.2 on a set X (briefly, 7.1C7.2) if 7.1 cl U C T2 cl U for every set U c 7.1 [258]. From this definition we immediately find that if 7.1 - co 7"1, then 7"1 is coupled to every topology on X so that the antidiscrete topology on X as well as the discrete topology on X is coupled to every topology on X. R e m a r k 2.2.2. By [258], if T1 is coupled to z2 on X, then T1 is coupled to every topology on X smaller than 7.2. But it is possible for a topology to be coupled to a strictly larger topology and in that case the coupling is mutual. For example, the antidiscrete topology is mutually coupled to every topology on the same set. The possibility for a topology to be coupled with a strictly larger topology leads the author of [258] to the notion of partial order. In [258] more interest is shown in the coupling of topologies than in the situation T1 el U C 7"2 el U for every set U E 7.2 (in our terms - the N-relation). One might think that this preference is conditioned by the following reasoning: the C-relation defines the partial order _< (in our notation < c ) on the family of all topologies on X by virtue of the equivalence T1%C 7"2 ~ (T1CT2 A T1 C T2), and in the subsequent investigations the author considers the cases where 7"1%C 7"2 and (X, 7.2) satisfies the conditions for which it is regular. If instead of the partial order < c we shall consider the relation
2.2. T h e C - R e l a t i o n
73
in [258], the conditions rl < x r2 and (X, r2) is regular (where the regularity of (X, r2) is not superfluous) imply that rl = r2. As distinct from the above situation, we have E x a m p l e 2.2.3. Let X = {a,b,c}, T 1 = {fg, {a}, {b, c } , X } , and r2 be the discrete topology on X. Then rlCr2 since 7"1 = COT1, T1 C T2, that is, rl < c r2 and (X, r2) is regular. However T 1 r T 2. The example below is given in the context of the above reasoning. E x a m p l e 2.2.4. Let X = {a, b, c, d, e}, T 1 = {f~, {a}, {b, c, d, e}, X}, and r2 = {;g, {a}, {a,c}, {a,b,c}, {a,c,d}, { a , b , c , d } , X } . Then rlCr2 and (X,~q,r2) is (2, 1)-quasi regular. However, rl and r2 are not comparable. The coincidence of the topologies for rl < c r2 demands a stronger requirement on (X, rl, r2), which is given below in Corollary 2.2.9. Note that following Definition 2.2.1, T 1 < C 7-2 ~ ( T I C T 2 /N T 2 C T 1 ) , but the inverse implication is not correct. E x a m p l e 2.2.5. Let X : {a, b, c, d}, T1 - - {~, {a}, {b, c, d}, X}, and r2 = {~, {b}, {a, c, d } , X } . Then T I C T 2 and T2CT1, but T1 and 7-2 a r e not comparable by inclusion. T h e o r e m 2.2.6. The following conditions are equivalent for a BS (X, T1,7-2): (1) 7"1 is coupled to r2. (2) 71 cl 71 int A c r2 cl T 1 int A so that r2 int
T1
cl A c
T1
int
T1
cl A
for every subset A C X . (3) vl cl r l i n t A c 72 cl A so that r2 int A c 7-1 int rl cl A
for every subset A c X . (4)
T1
sup(7-1~7"2)"
(5) For every point x E X the 1-closure of any 2-neighborhood of x is a 1-neighborhood of x.
P oof.
quiv l nr
(1) r
(4)
(5)
p ov d in Theorems 2
5 of [258]. (1) ~ (2) is obvious by Definition 2.2.1. If (3) is satisfied, then r l c l d c T2clm for every 1-open set d so that (3) ~ (1), where (1) ~ (2). Since the implication (2) ~ (3) is evident, we obtain (2) ~ (3). 77 A set X together with topologies rlCr2 (rl < c r2) is denoted by (X, rlCr2) ( ( X , T1 < C
T2)).
C o r o l l a r y 2.2.7. The conditions below are equivalent for a BS (X, rl,r2): (1)
T1
7"2"
(2)
T 1 C1 T 1
int A
= T2
cl T 1 int A so that rl int T1 cl A = r2 int
for every subset A c X .
T1 c l A
74
II. Different Relations Between Two Topologies . . . (3)
nl(U)
~ for
-
each set U ~ 7.1 8o that
n 2 ( F ) - 2~ for each set F c co7.1. (4) l-C17.1
--
2-C1 T1 80 that l-Int co 7.1 -- 2-Int co 7.1.
Proof. The equivalences are immediate consequences of Definitions 1.4.19 and 2.2.1, and (2) of Theorem 2.2.6, taking into account the inclusion T 1 C T 2. [-7 C o r o l l a r y 2.2.8. The conditions below are satisfied for a BS (X, 71C7.2)" (1) If (2,7-1,7-2) i8 (1,2)-regular ((1,2)-quasi regular), then (X, 7.1,T2) i8 1-regular (1-quasi regular), and if (X, 7-1, "/-2) i8 1-connected, then (X, T1, T2) is p-connected. (2) If (Y, 7.~, 7.~) is a BsS of X and Y c T1 U T2, then 7.~C7~. Moreover, for a BS (X,
T1 < C
7"2), we have"
(3) (X, 7.1,7.2) is (1, 2)-regular ((1, 2)-quasi regular) .z---->. (X, 7.1,7.2) is 1-regular (1-quasi regular) and (X, 7.1,7.2) is 1-connected .z--->. (X, 7.1, 7.2) is p-connected. (4) If (X, 7.1,7.2) is p-normal, then (X, 7.1,7.2) is 1-normal. (5) If (X, 7.1,7.2)is a l-T1 BS, then X ~ - X~ and, in this case, if 7.2 is the discrete topology, then 7"1 --7.2. (6) 7.1 n 1-D(X) - 7.1 n 2-D(X) so that 9
CO 7.1 n 1 - ~ d ( X )
(7)
1 - I n t e l 7.1 -
-
c o 7.1 (~ 2 - ~ d ( X ) .
2-Int C17.1 - (1, 2)-Int C171
--
(2,
1)-Int C1T1,
so that 1-C11nt co 7-1 - - 2-C11nt co 7.1 - -
(1, 2)
-C11nt co 7.1 - (2, 1) -C11nt co 7.1-
(X, 7.1,7.2) is p-extremally disconnected ~ (X, 7.1,7.2) is 1-extremally disconnected. (9) A E (2, 1)-N'D(X) if and only if for every set U E 7.2 such that 7.1 int U r ~, there exists a set V c 7.1 \ { ;2~} together with V c U and V N A - ;g. (10) If (Y, 7.~, 7.~)is a BsS of X and Y E 2-79(X) then T~
Proof. (1) The first part is an immediate consequence of (2) of Proposition 0.1.7 and Definition 0.1.14, taking into account Definition 2.2.1. Now, suppose that (X, 7.1,7.2) is not p-connected. Then there is a set U E 7.1 f? co 7.2 such that 2~ :/: U r X. Then U c 7.1 and 7.1C7.2 imply that 7.1 C1U C 7.2 cl U - U. Hence U E T1 ('l CO 7.1 SO that (X, 7.1,7.2) is not l-connected. (2) Begin by assuming that Y c 7.1 and U c 7.~ is any set. Then U E 7.1 and by Definition 2.2.1, 7.1 cl U c 7.2 cl U. Hence 7.1 cl U n Y C 7.2 cl U 5 Y so that 7.1' c l U C 7.2' cl U and so 7.1'C7.~ 9 If Y c T2 and A C Y is any set, then by (3) of Theorem 2.2.6, T2 int A C
T1
int
7-1 cl
A
2.2. The C-Relation
75
and so z~ int A = 7.2 int A c T1 int 7-1 cl A • Y c T~ i n t (7.1 cl A A Y) = 7.~ int 7.~ cl A. Thus 7.1'Cry. (3) The first part follows directly from the first part of (1), taking into account (3) of Corollary 2.2.7. The second part is an immediate consequence of the second part of (1) and the implications after Definition 0.1.18. (4) Let F c co7.1, U c T1 and F C U. Then F c co7.2 and by (4) of Proposition 0.1.7, there is a set V c 7.1 such that F C V c 7.2 cl V C U. Hence, by (3) of Corollary 2.2.7, we have F c V c T1 cl V C U and so (X, 7.1,7.2) is 1-normal. (5) The first part follows immediately from (3) of Corollary 2.2.7. The rest is obvious. Assertions of (6) are also consequences of (3) of Corollary 2.2.7. (7) If U c TI~ then by (3) of Corollary 2.2.7, we have T 1 C1U -- 7.2 c 1 U . Hence T1
int 7.1 cl U = 7.1 int 7.2 cl U,
7.1 int 7.1 cl U = 72 int 7.1 C1U -- 7.2 int 7.2 cl U
and thus 7"1 i n t 7.1 cl U -- 7.2
int 7.2 cl U = 7.1 int r2 cl U = r2 int rl cl U.
(8) By ( 3 ) o f Corollary 2.2.7 and (7) above, we have (7.1 cl 7.2
int 7"1 C1A ~- 7-2 int 7.1 C1 A ) ~
(7.1 c1 7.1
int 7.1 cl A - 7.1 int 7"1 C1 A )
for every subset A c X. Hence it remains to use (3) of Definition 0.1.18 and the equivalences after Definition 0.1.18. (9) The necessity follows directly from Corollary 1.1.6. On the other hand, if 7.2 int 7.1 cl A r 2~, then by (2) of Lemma 0.2.1 and (2) of Corollary 2.2.7, we have 7.1 int 7.2 int 7.1 cl A = T1 int 7.1 cl A = 7.2 int 7.1 cl A r 2~. If V c 7.1 \ { ~ } and V c 72 int 7-1 el A , then V c 7.1 cl A so t h a t V ~ 7.1 cl A = V r and thus V C~A r ~. (10) Finally, let Y c 2-~D(X) and g ' c 7.{ be any set. Then, there is a set U c 7.1 such t h a t U ~ = U A Y . By (3) of Corollary 2.2.7, 7.1clU = 7.2clU and Y c 2-Z)(X) c 1-Z)(X) implies t h a t T1
cl U = T1 cl(U c~ Y) = 7.2 c1 g = 7.2 c l ( g ~ Y ) .
Hence 7.~ cl U ~ - 7.~ cl U t and it remains to apply once more (3) of Corollary 2.2.7. F-1 C o r o l l a r y 2.2.9. If (X, TICT2) i8 a (2,1)-regular BS, then 7.2 c 7.1 and, therefore, for a (2, 1)-regular BS (X, 7.1 < c 7.2), we have 7.1 = T2.
Proof. Let U E 7-2 be any set (2,1)-regularity of X, there exists But, according to (5) of Theorem there exists a set W E T1 together x r U being arbitrary implies t h a t
and x c U be an arbitrary point. By the a set V r 7.2 such that x c V c 7"1 cl V C U. 2.2.6, 7"1 C1V is a l-neighborhood of x so that, with x ~ W c T1 C1 V . Hence x E W c U and U c 7"1. [--]
Example 2.2.4 shows that the requirement that (X, 7.1,7-2) be (2,1)-regular in the last corollary is not superfluous.
76
II. Different Relations Between Two Topologies ...
C o r o l l a r y 2.2.10. If (X, 7"1C7"2) is a (2, 1)-quasi regular BS, then for every set U c 7"2 \ {~}, there is a set V c 7"1 \ {~} such that 7"1clV c U.
Proof. It is clear that for every set U c 7"2\{2~} there is a set W c 7"2\{2~} such that 7"1cl W c U. Let x E W be any point. Then by (5) of Theorem 2.2.6, 7"1 cl W is a 1-neighborhood of x and, therefore, there is a set V c 7"1 \ { ~ } s u c h that x c V c 7"1 cl W. Thus 7"1 cl V c U. [3 D e f i n i t i o n 2.2.11. A topology 7"1 is No-coupled to a topology 7"2 on a set X (briefly, 7"1C(R0)7"2) if 7"1 is coupled to 7"2 and 17"1int 7"1cl A \ 7"2int A I < No for every subset A c X. Clearly, by (2) of Corollary 2.2.8, the C(N0)-relation between topologies rl and r2 on a set X is hereditary with respect to any set A c rl tOr2 and, moreover, by (10) of the same corollary, the
It is obvious that if IX[ _< N0, then rlCr2 ,<---->tiC(No)r2. However, we have E x a m p l e 2.2.12. Let X be an uncountable set, ~ r A c X, [A I < N0, A, X \ A, X}, r2 = {~, X}. Then T1CT2, but T1 is not R0-coupled to r2 since for the set X \ A, we have
T 1 -- { ~ ,
Irl int rl cl(X \ A) \ r2 int(X \ A)[ - IX \ A I > N0. The C-relation is not hereditary with respect to/-closed subsets in general. E x a m p l e 2.2.13. Let (X, 7-1,7-2) be the BS from Example 2.1.8. Then T2. Let us consider the set F = {c, d} c co T1 C CO 7-2. Then for the subset {d} of a BsS (Y, r~, r~), we have r~ int r~ cl{d}=2~ and r~ int{d}={d}, that is, r~ is not coupled to r~.
T1 ~ C
D e f i n i t i o n 2.2.14. A topology 7-1 is/-strongly coupled to a topology 7-2 on a set X (briefly, rlC(i)7-2) if the C-relation is hereditary with respect to/-closed subsets of X. E x a m p l e 2.2.15. Let X = {a, b, c, d}, rl = {2~, {a, b, c}, X } and 7-2 = {2~, {a, b}, {a, b, c}, X}. Then T1
is a
1-T1 BS, then
co r2 • 1-DZ(X) = 2-P(X).
Proof. Let F c co r2 Cq 1-DZ(X). Following the condition, r~ < c r~ for (F, r;, r~) and by (5) of Corollary 2.2.8, F~ = F~ = 2~ since F c 1-DZ(X). Hence F c F~, that is, F E 2-DZ(X) and so F c co
n 2-DZ(X) = 2-P(X).
2.2. The C-Relation
77
Thus co T2 N 1-/)Z(X) C 2-7)(X). [-1
The reverse inclusion follows from (2) of Proposition 1.4.2. Note that
(T1C(i)T2 A T1C(}{O)T2) ~
TIC(i, Ro)T2 ~
T;C(Ro)T;
and (T1 "~C(i)7"2 A T1C(~o)T2) ~
T1
T~
for every/-closed BsS (F, T[, T~) of a BS (X, 71,72). Using the p-quasi regularity argument, we obtain the following important statement, interconnecting the C- and S-relations. T h e o r e m 2.2.17. Let (X, T1, T2) be a p-quasi regular BS. Then (7-1CT2 A T2CT1) ~
T1ST2.
Proof. By virtue of (2) of Theorem 2.1.5, it suffices to show that the equivalence T1 int A -r ~ ~ 72 int A -r 2~ holds for any subset A c X. Begin by assuming that T1 int A =/= 2~. Then by the (1,2)-quasi regularity of (X, 71, r2), there exists a set V c T1 \ { ~ } such that 72 cl V c 7-1 int A. If x c V is any point, then by (5) of Theorem 2.2.6, there is a 2-open neighborhood U(x) such that U(x) c r2 el V as T2C~-I. Hence U(x) c T1 int A so that 72 int T1 int A -r 2~ and thus T2 int A =/= ~. Using the (2,1)-quasi regularity of (X, 71, T2) and TICT2 one can prove similarly that T2 int A -r 2~ implies T1 int A =/= 2~. [El C o r o l l a r y 2.2.18. Let (X, 7-1,72) be a p-quasi regular BS, where T1CT-2 f T2CT1. Then (X, T1,72) is a 1-Baire space if and only if (X, 7-1, T2) is a 2-Baire space.
Proof. Under the hypotheses, we have T1ST2 and it remains to use Remark 2.1.3. [-1 C o r o l l a r y 2.2.19. If (X, T1
Pro@ Indeed, it is clear that 7"1
9-1CT2):
(1) I-N'Z?(X) c (2, 1)-Af~(X) so that (2, 1)-SZ)(X) c 1-S~D(X) and thus
1-Ca~gi(X ) ~ (2, 1)-Cai~gi(X), (2, 1)-Catgii(X ) c 1-CaI~gii(X ).
~
-
o
~
II
_
~
~,.
Y
~
.
~
~
il
Rg
~
~-
~
~
~
~
~
~
~
~
~
o
~
~
~
0
~
~
,_,~
9
,
0
~
~
oo
~
~"
.~
~-'~
i-,o
~
~
8 ~
~." ~"
o
~ ~ ~. . ~
~
~
c~
~
~
q
n
~
,~
" T ' ~
n
--
~
~
_~
~
"T"
~
~
~~
~
~
~
C~
II
-.
6~ ~~ ~~
II
~
~
II
,-'
~
~
,~ ~
~
~
.-.
ii
~ ~
~
~
>~
II
~
~
~
~ ~~ ~
~
'
,~
"r"
~
~
~
~
~3
~
~
r~
:b_,~
II
~-.
~
~
'~
-"
~
"~ - - ~
,
9
~.
c"'b
2.2. T h e C - R e l a t i o n
79
and so
~-eD(x)
= (~, 2 ) - e D ( x )
c 2-eD(x).
Furthermore, (2) of Corollary 1.3.7 also gives
~-~ n 2 - e ~ ( x )
=
(1,
2)-eD(x) n 2-oD(x)
= (1,
2)-eD(x)
and, therefore, T1 ["] 2 - O ~ ) ( X )
----- 1-OD(X)
= (2, 1 ) - O D ( X ) - (1,2)-OD(X).
The second condition in (4) is equivalent to the first one. Finally, according to Definition 1.3.26, (1) of T h e o r e m 1.3.27 and the first condition in (4),
A c 1-SOZ)(x)=
1-scz)(x)<
;
<---> (there exists a set U E 1-OD(X) such t h a t U c A c r~ cl U) < ;e, > (there exists a set U c (2, 1)-OD(X) such t h a t U c A c rl cl U) e, > > A ~ (2, 1)-SOD(X) = (1,2)-,5CD(X).
e
Moreover, Definition 1.3.26, (1) of T h e o r e m 1.3.27, the first condition of (4) above and (3) of Corollary 2.2.7 imply t h a t
A c (1,2)-soz)(x)=
( 2 , 1 ) - s c z ) ( x ) ,,, ;
,,' > (there exists a set U c (1, 2)-OD(X) such t h a t U < A < r2 cl U) -,' ;< ;, (there exists a set U c 1-OD(X) such t h a t U c A c r2 cl U = rl cl U) e, ,,
e ;, A < 1-,SOD(X)= 1-SCD(X). Thus, we have
~-soz)(x)
= ~ - s c z ) ( x ) : (~, 2 ) - s o y ( x ) = - (2,1)-soz)(x)
(2,1)-acz)(x) :
= (~, 9 ) - s c z ) ( x ) .
It is also clear t h a t A ~ (1,2)-Soz)(x)=
(2,1)-scz)(x)
~, ,,,
e, ;, (there exists a set U c (1, 2)-O79(X) such t h a t U c A c r2 cl U) < > e, > (there exists a set U E T1 ("12-079(X) such t h a t U C A c r2 cl U) < ?-
e, ,, A < 2-,...qO~)(X)/T1, where the last equivalence follows from Definition 1.3.33. The rest is obvious by Proposition 1.3.34.
D
Note t h a t for the reverse inclusion 2-OD(X) c 1-OD(X), the requirement in (4) of T h e o r e m 2.2.20 for 2-open domains to be 1-open is not superfluous. Example
2 . 2 . 2 1 . Let (X, rl, r2) be the BS from E x a m p l e 2.2.3. T h e n {a, b} c
2-o7)(x), but {a, b}-g 1-OD(X) : (2, 1)-OD(X) = (1, 2)-OD(X) since { a, b} g rl. Corollary
2.2.22.
The following implication holds for a BS (X, rl < c r2):
m
II ~ m
~ -. ~
7-.
& ~ &
II
~
"7"
.,
,
,~, .. ~
7-.
,-~
.,
,~ ,";-" c ~ ~
~
~ ~
7_.
~
m
, ~ ,
m
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2.2. The C-Relation
81
(5) A c (1, 2 ) - $ O ( Y ) -
1-SO(Y) ~
A c 2-80(Y) ~
A c (1, 2 ) - S O ( X ) -
1-SO(X) ~
A c 2 - S O ( X ) ==~ A c (2, 1)-SO(X).
I f Y C T1 n CO 7"1,
A c (2, 1)-SO(Y)
then
(6) A c ( 1 , 2 ) - S C ( Y ) - 1 - S C ( Y ) ~ A E (1, 2 ) - S C ( X ) -
1-SC(X) ~
A c 2 - S C ( Y ) -~. A c (2, 1 ) - S C ( Y )
A c 2-SC(X) ~
A ~ (2, 1)-SC(X).
If Y e (1, 2 ) - O D ( X ) , then
(7) A E (1, 2 ) - O D ( Y ) - 1 - O D ( Y ) -
(2, 1 ) - O D ( Y ) - 2 - O D ( Y ) N 7-;
A E (1, 2 ) - O D ( X ) - 1 - O D ( X ) - (2, 1 ) - O D ( X ) - 2 - O D ( X ) n T1, and thus A c (1,2)-SOD(Y)
- ( 2 , 1)-,SCD(Y)- 1 - S O D ( Y ) - 1-,SCD(Y) - ( 2 , 1 ) - S O D ( Y ) - (1,2)-SCD(Y) - 2-SOD(Y)/'r; = -
A c (1,2)-,SOD(X)
If Y c
2-scz
(Y)/co
- ( 2 , 1 ) - S C D ( X ) - 1 - S O D ( X ) - 1-$CD(X) - ( 2 , 1)-SOD(X) - ( 1 , 2)-$CD(X) - 2 - S O D ( X ) / T 1 = 2-SCTP(X) / c o "q.
--
n (1, 2 ) - c D ( x ) ,
(8) d c (1, 2)-CD(Y) - 1-CD(Y) - (2, 1)-riD(Y) - 2-CD(Y)N co'r~ A c ( 1 , 2 ) - C D ( X ) = 1 - C D ( X ) = (2, 1 ) - C D ( X ) = 2 - C D ( X ) N co T1. Proof. Note that by (2) and (10) of Corollary 2.2.8, in all cases (1)-(8) the C-relation, as well as the
82
II. Different Relations Between Two Topologies ...
(6) It is obvious that (5) of Theorem 2.2.20 also gives the horizontal equalities and implications in this condition. The vertical equivalences follow directly from Corollary 1.5.30 and its topological version. (7) The horizontal equalities of the upper condition are given by (4) of Theorem 2.2.20. Corollary 1.5.22 implies the vertical equivalence. The horizontal equalities of the lower condition are also given by (4) of Theorem 2.2.20 and the vertical equivalence follows from Corollary 1.5.32. (8) By (4) of Theorem 2.2.20 the horizontal equalities are obvious. According to Proposition 1.5.23 if A E (1,2)-CD(X), then A E (1, 2)- CD(Y). On the other hand, Y E (1,2)-CD(X)= (2, 1)- CD(X) and Proposition 1.5.25 imply that (2, 1)-CD(Y) = (1, 2)-CD(Y) C (2, 1)-CD(X) = (1, 2)-CD(X). Thus the vertical equivalence holds.
D
P r o p o s i t i o n 2.2.24. The following conditions hold for a BS (X, 7"1 < c 7"2): (1) X is of 1-CatglI(
II --5, 1-~5(X) N 2-D(X) c 2-~;5(X) N 2-D(X)
N
N
1-Ga(X) ~ 1 - D ( X ) c 2-G5(X)A 1- D ( X ) c
C (1,2)-CatgII(X) C 2-CatglI(X) C 1 - C a t g I I ( X ) = (2,1)-Catgxi(X).
Proof. Assertions of (1) follow directly from the inclusion 2-D(X) c 1-D(X), taking into account (3) of Theorem 2.2.20. The proof of (2) follows immediatelly from Corollary 1.1.27 and (3) of Theorem 2.2.20. 2.3. T h e N - R e l a t i o n In Remark 2.2.2 we have mentioned that the relation existing between two topologies 7-1 and 7"2 on a set X and defined as 7"1 C1 U C 72 c1 U for every set U c 7"2 is not considered in [258] because of the coincidence of topologies 7"1 and 7"2 for a 2-regular BS (X, T 1 < N 7"2). Nevertheless, based on Corollary 2.2.9, we have a full right to consider the relation 7"1
7"1N7"2) if 7"1 c1 U c 72 c1 U for every set U r 72. E x a m p l e 2.3.2. Let (]t~,~l,Cd2) be the natural BS. S-related to Cdj n o r coupled to c~j; however, wi is near a~j.
Then ca~ is neither
From Definition 2.3.1, we readily derive the following simple properties of nearness. P r o p o s i t i o n 2.3.3. The following conditions are satisfied for a BS (X, 7"1,7"2): (1) (7"1N7"2 A 7" C 7"2) : = ~ 7"1N7". (2) (7"1N7"2 A 71 C 7") ~ 7"N7"2.
2.3. The N-Relation (3) (7"2 C 7"1 V 7"2 C CO 7"1) ~
83
7"1NT"2-
Proof. (1) It is clear t h a t for each set U c 7" c 7"2, we have 7"1 C1U C 7"2 CI U C 7" CI U so t h a t 7-1N7.. (2) and (3) are obvious. [~ C o r o l l a r y 2.3.4. The discrete topology on X is near any topology on X and every topology on X is near the antidiscrete topology on X . In contrast to the S-relation, the N-relation, like the C-relation, is not symmetric. E x a m p l e 2.3.5. Let X = {a, b, c}, 7"1 = { ~ , {a, b}, X } , &nd 7"2 = { ~ , {a}, {b}, {a, b}, {b, c}, X } . T h e n 7"2N7"1 since 7"1 C 7"2, but 7"1 is not near 7"2 since for the set {b, c} E 7"2, we h&ve 7"1 cl{b, c} -~ X &nd 7"2 cl{b, c} -- {b, c}. It is obvious t h a t 7"1 ~ N 7"2 ~ Example
X\{a,b}
2.3.6.
:/: ~.
IfT"1
(7"1N7"2 A 7"1 C 7"2).
Let X be any set such t h a t a, b c X, a -~ b, implies t h a t =
{~,{ai, X\{a},X},
7"2 = { ~ , { a } , X \ { a } , { b } , { a , b } , X } ,
then 7"1 < N 7"2Theorem
2.3.7.
The following conditions are equivalent for a BS (X, 7.1,7.2):
(1) 7-1 is near T2. (2) 7-1 cl 7"2 int A C 7"2 cl 7"2 int A so that 7"2 int 7"2 cl A c 7"1 int 7"2 cl A
for every subset A c X . (3) 7"1 C1 7"2 int A C 7"2 cl A so that 7"2 int A c 7"1 int 7"2 cl A
for every subset A c X . (4) For every point x c X l-neighborhood of x.
the 2-closure of any 2-neighborhood of x is a
Proof. It is obvious t h a t (1) ~ (2) ==~ (3). If (3) is satisfied, then 7"1CI U C 7"2 cl U for every set U c 7"2 and so (3) ==~ (1). (1) ==~ (4) Let x c X be any point and U(x) be its any 2-neighborhood. W i t h o u t loss of generality it can be assumed t h a t U(x) E 7"2. T h e n V = X \ ~2 cl U(x) c ~2
and by (1), 7"1 cl V c 7"2 cl V. If V1 = X \ 7"1 C1 V, then Pl -- X \ 7"1 cl ( X \ 7"2 cl U ( x ) ) - 7"1 int 7"2 cl U(x) C 7"2 c1U(x).
x c
cl U(x)
\
cl
U(x) =
7"2 int ( X \ U(x)) N U(x) - ~, we obtain t h a t 7"2 cl 7"2 int ( X \ U(x)) n U(x) - ~.
int(X \
Sinc
84
II. Different Relations Between Two Topologies ...
Hence x ~ v2 cl (X \ ~-2 cl U (x)) - T2 cl V and T1 cl V c T2 cl V implies x ~ T1 cl V so t h a t x E V1 - X \ T1 cl V. Thus V1 is a 1-open neighborhood of x such t h a t V1 C v2 cl U(x). (4) ~ (1) Suppose t h a t U E T2 is any set and x E T1 cl U is an arbitrary point. If U(x) is any 2-open neighborhood of x, then by (4), there exists a set V c TI such t h a t x E V c ~-2 cl U(x). It is clear t h a t V N U --~ ~ so t h a t v2 cl U ( x ) N U ~ ~. Hence U(x) A U ~ ~ and U(x) c 7-2 being arbitrary implies t h a t x E ~-2 cl U. D Remark
2.3.8. By analogy with (4) of Theorem 2.2.6, we have (T1 < N sup(T1,7-2)) ~
TINT2.
Indeed, if T1
Corollary 2.3.9.
The relation
topologies on a set X . Pro@ Clearly, it suffices to show only t h a t (rl < N 3-2 A 3-2 < N 7-3) ~ T1 < N 7-3. Suppose t h a t x c X is any point and U(x) c r3 is its any neighborhood. Since r2 < g ra, following (4) of T h e o r e m 2.3.7, there exists a neighborhood W ( x ) c r2 such t h a t W ( x ) c 7-3 cl U(x). The condition rl
[--]
Examples 2.2.5 and 2.3.2 show t h a t the notions of coupling and nearness of topologies are independent of each other. However, we have the following obvious
Corollary 2.3.10. The implication 7-1
~ N 7-2 ~
7-1 ~ C 7-2 is correct for a
BS (X, 7-1,72).
E x a m p l e 2.3.11. Let X - {a,b,c}, T 1 be the antidiscrete topology on X, 7-2 -- { ~ , {a}, {b, c } , X } . T h e n T1 ~ C T2, b u t T1 ~ N T2 is n o t correct.
Corollary 2.3.12. The conditions below are equivalent for a BS (X, 7"1, T2)" (]) T1 ~ N 7-2. (2) T1 C17-2 int A - ~-2 cl ~-2 int A so that 7"1 int 7-2 cl A - ~-2 int T2 cl A
for every subset A c X . (3) n l(U) - ~ for each set U E T2 so that n 2 ( F ) - ~ for each set F c coT2.
2.3. T h e N - R e l a t i o n
85
(4) l-C1 r2 = 2-C1 r2 so that 1-Int co r2 = 2-Int co r2.
Proof. T h e p r o o f is b a s e d on Definitions 1.4.19 a n d 2.3.1 in c o n j u n c t i o n w i t h (2) of T h e o r e m 2.3.7 a n d t h e inclusion T1 C 7-2. [--] Corollary
2.3.13.
The following conditions are satisfied for a BS (X, ~-INT2):
(1) If (X, rl, r2) is 2-regular (2-quasi regular), then (X, rl, r2) is (2, 1)-regular
2-connected. (2) If (Y, r~, ~;) is a BsS of X and Y c ~ 0 2-Z~(X), then rr
Moreover, for a BS (X, rl < N r2), we have: (a) ( x , (1, (1, 2)-q a i
2-q a i
regular), and (X,rl,r2) is I-connected <--> (X, rl, r2) is p-connected ~' ". (X, rl, r2) is 2-connected .e----->.(X, rl, r2) is d-connected. (4) If (X, rl, r2) is 2-normal, then (X, rl, r2) is p-normal and so (X, rl, r2) is 1-normal. (5) r2 A 1 - D ( X ) = r2 A 2 - D ( X ) so that
co (6)
= co
1-Int C1 r2 - 2-Int C1 r2 - (1, 2)-Int C1 r2 - (2, 1)-Int C1 r2
so that 1-C11nt co r2 - 2-C11nt co 72 - (1, 2)-C1 Int co r2 - (2, 1) -C11nt co r2. (7) (X, r l , 72) is p-extremally disconnected e----->, (X, 71~ 7-2) i8 2-extremally disconnected <---> (X, 71, r2) is 1-extremally disconnected e----> (X, r l , r2)
is d-eztremally disconnected. Proof. (1) T h e first p a r t is an i m m e d i a t e c o n s e q u e n c e of (2) of P r o p o s i t i o n 0.1.7 a n d Definition 0.1.14 in c o n j u n c t i o n w i t h Definition 2.3.1. If (X, rl, r2) is not 2-connected, t h e n t h e r e is a set U c r2 N co r2, ~ :/: U ~- X a n d rlNr2 implies t h a t T1 cl U c ~-2 cl U - U, a n d so U c r2 c~ co ~-1. Similarly, one can prove t h a t X \ U c r2 ca COrl. H e n c e U c rl N c o r l so t h a t (X, r l , r2) is n o t 1-connected. (2) A s s u m e t h a t Y c r2 a n d A c Y is any subset. T h e n by (3) of T h e o r e m 2.3.7, r2 int A c rl int r2 cl A so t h a t r 2' int A - r2 int A c rl int 72 cl A cq Y c !
T 1'
int(r2 cl A Cq Y) -
T 1'
int r~ cl A
!
a n d so T1 N % . O n t h e o t h e r h a n d , let Y c 2 - D ( X ) a n d A c Y be a n y set. Clearly, t h e r e exists a 2-open set U such t h a t U cq Y - r~ int A. B y (3) of T h e o r e m 2.3.7, U c r l i n t r2 cl U a n d hence %' int A C rl int r2 cl U N Y c T1' int(r2 cl U cq Y) - r~ int (r2 cl(U cq Y) N Y) -
-
T 1' i n t
(r2 cl rs int A Cl Y) c r~ int r~ cl A
since Y E 2 - D ( X ) , a n d so r ~ N r ~ .
86
II. Different Relations Between Two Topologies ...
(3) The first part follows directly from (3) of Corollary 2.2.8, the first part of (1) above, and (3) of Corollary 2.3.12, taking into account Corollary 2.3.10. The second part is an immediate consequence of the second part of (1) and the implications following Definition 0.1.18. (4) By (4) of Corollary 2.2.8 and Corollary 2.3.10, it suffices to show only the correctness of the first implication. Let F c co7-1, U c 7-2 and F C U. Since F c co7-1 C co7-2 and (X, 7-1,7-2) is 2-normal, there is a set V c 7-2 such that F c V c 7-2 cl V c U. Hence, by (3) of Corollary 2.3.12, F C V C 7-1 cl V C U and it remains to use (4) of Proposition 0.1.7. Assertions of (5) follow immediately fl'om (3) of Corollary 2.3.12. (6) If U E 7-2, then this proof is identical to that of (7) of Corollary 2.2.8, taking into account (3) of Corollary 2.3.12. (7) The proof of the first equivalence is similar to the proof of (8) of Corollary 2.2.8, taking into account (6) above and (3) of Corollary 2.3.12. The rest follows from (8) of Corollary 2.2.8 in conjunction with Corollary 2.3.10. K] Thus, by (1) of Corollary 2.3.13, the conjunction (7-1N7-2 A 7-2N7-1) implies the equivalence (X, 7-1,72) is 1-connected ,+-->, (X, 7-1,7-2) is 2-connected.
D e f i n i t i o n 2.3.14. A topology 7-1 is R0-near a topology ~-2 on a set X (briefly, near 7-2 and 17-1int 7-2 clA \ 7-2 int A I < Ro for every set A c X.
TIN(Ro)T2) if 7-1 is
It is obvious that if IXI < Ro, then 7-1N7-2 ~ 7-1N(Ro)7-2. However, ~iN~j, but ~i is not Ro-near ~j in the natural BS (R, 021, ~22). Clearly, by (2) of Corollary 2.3.13, the N(R0)-relation as well as the
(71N(~0)7-2/~ 71 C 7-2).
T1
As for the C-relation, Example 2.1.8 also shows that the N-relation is not, in general, hereditary with respect to/-closed subsets. D e f i n i t i o n 2.3.15. A topology 7-1 is/-strongly near a topology 72 on a set X (briefly, 71N(i)7-2) if the N-relation is hereditary with respect to/-closed subsets of X. It is obvious that in Example 2.2.15 we also have 7-1
,
,
'X(
o)4
and (T1
71
T;
for every/-closed BsS (F, 7-~,7-~) of a BS (X, T1, T2). Proceeding with our investigation, note that by Example 2.3.2, the N- and S-relations are independent of each other. Moreover, we have
2.3. The N-Relation
87
E x a m p l e 2.3.16. Let X = {a,b,c,d}, 771 = {25, {a, bI, X } , and 772 = {2~, {b}, {a,b}, { b , c , d } , X } . Then 771
2.3.17. The statements below hold for a BS (X, 711 772):
(1) If (X, 7711772) is d-quasi regular, then (771 N772/k 7-2NT1) ===~71 S772.
(2) If (X, 7711772) is p-extremally disconnected, then 7-1 $772 -----N, (771N7-2/k 772ST1).
Proof. (1) By virtue of (2) of Theorem 2.1.5, it suffices to show t h a t rl int A =/= ~ <---> r2 int A =/= ~ for every subset A c X. If rl int A r 2~, then by the 1-quasi regularity of (X, rl, r2), there is a set V c rl \ {~} such t h a t T1 C1 V C rl int A. Let x c V be any point. Then, by (4) of Theorem 2.3.7, there exists a 2-open neighborhood U(x) such that U(x) c 7-1 C1V since 772N771. Thus U(x) c rl int A and so 2~ :fi 72 int T 1 int A C r2 int A. The implication T2 int A -r 2~ ----5. 771int A -r 2~ can be proved in a similar way using the 2-quasi regularity of (X, 771,772) and TINr2. (2) Following (3) of Theorem 2.1.5, T1 ST2 ~
(T1 int A C T1 C1T2 int A A T2 int A c 72 el T1 int A)
for every subset A C X. Since (X, 71, r2) is p-extremally disconnected, by (3) of Definition 0.1.18, 7-2 CI T 1 int A = 771int 772cl T1 int A A T1 C1 7-2 int A = 72 int T 1 C1 7-2 int A
for every subset A c X. Hence 72 int A c 771int T2 cl T 1 int A c 771int 772el A and T1 int A c 772int T1 C17"2 int A c 72 int 771cl A
for every subset A c X. Thus it remains to use (3) of Theorem 2.3.7.
[-7
C o r o l l a r y 2.3.18. The following conditions are satisfied for a BS (X, 771,T2):
(1) If ( X, 771 < x 772) is d-quasi regular, then 771 < s 772. (2) If (X, 771,7-2) is d-quasi regular, where TINTT2 A 7"2NTT1, then (X, 771,7"2) i8 1-Baire if and only if ( X 1771172) i8 2-Baire. Moreover~ ( X 17711772) i8 p-quasi regular. (3) If (X, T1 < s 72) is p-extremally disconnected, then 771
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2.3. T h e N - R e l a t i o n
(9) A ~ (1, 2)-YD(Y) = 2 - H ~ t Y ) ~ so that A c (1, 2 ) - S T ) ( Y ) = 2-$T)(Y) ~
91
A ~ 2-MD(X) = (1, 2)-HD(X) A c 2 - $ T ) ( X ) = (1, 2)-$T)(X)
and thus A c (1, 2)-Catg I ( Y ) - 2-Catg I (Y) ~
A c 2-Catg I ( X ) - (1, 2)-Catg I (X),
so that A c (1,2)-Catg~i(Y)=2-Catgix(Y) ,z--~, A c 2-Catgii(X)=(1,2)-Catgxi(X).
(10) A c 2 - S O ( Y ) = (2, 1 ) - s o ( Y ) ~ I f Y C 7-1 n
c o T2,
A c (2, 1 ) - S O ( X ) =
2-SO(X).
then
(11) A E 2-SC(Y) = (2, 1)-SC(Y) ,e--->, A c 2 - S C ( X ) = (2, 1)-SO(X). Proof. In common with Corollary 2.2.23 we begin by observing that by (2) of Corollary 2.a.la in all cases (1)-(11) both the N-relation and the
, X is of 2-gary II) implies that
N
n
1-0a(X) N 1-D(X) C 2-0a(X)Cl 1-D(X) C (1, 2)-Catgii (X) 2-Catgli (X) c (2, 1)-Catgii (X) = 1-Catg~i (X).
----
Proof. The proof follows directly from (2) of Proposition 2.2.24, taking into account (3) of Theorem 2.3.19. D Now, we give the appropriate example from [14, p. 159, Exercise 230]. E x a m p l e 2.3.23. Let (X, rl < r2) be a 1-compact and 1-first countable BS, where IUI > R0 for each U E r l \ { ~ } and r2 = { U \ A : U c rl, A c X, IAI < R0}. Then rl
92
II. Different
Relations
Between
Two
Topologies
...
At the end of the section we introduce an equivalence relation D on a special proper subfamily B(T) C A ( r ) , where A ( r ) is the family of all topologies on a given set X, and we prove the following important T h e o r e m 2.3.24. Let X be any set and B(r) be any family of topologies on X , satisfying the following conditions: (1) B(T) is closed under finite intersections. (2) The antidiscrete topology ~-A - { ~ , X } ~ B ( ~ - ) .
Then there exists an equivalence relation D on B(T) such that if one member of an equivalence class is connected, then all members of the class are also connected. Proof. It is not difficult to see that a binary relation ~ on B(T), defined as follows: ~-1 ~ ~-2 if there is a topology ~- c B(T) such t h a t 7"1 C1U - - 7"2 cl U for each U c T, is an equivalence relation on B(T). Further, if we suppose that D is a binary relation on B(T), defined in a manner as follows" 71D72 ~ (71N72 A 72N71), then D is an equivalence relation in the above sense. Indeed, by Definition 2.3.1, TIDT2 ~ ((T1 cl U C_ T2cl U for each U c ~-2) A (T2 cl U c T1 cl U for each U c T1)), and since by (1), B(T) is closed under finite intersections, we obtain T1 CI V - - T 2 CI V for each V c ~ - - T1 N T2. Finally, it follows immediately from (1) of Corollary 2.3.13 t h a t if one member of a D-equivalence class is connected, then all members of this class are also connected. [3 2.4. B i t o p o l o g i c a l Insertions The notion of insertion property of a topology ~- on a set X was introduced in [173] to describe some classes of functions and spaces. In this section we shall define and investigate some bitopological modifications of this notion with an aim to apply them in characterizing the relations between two topologies as well as the small and large p-inductive dimension functions in the next chapter.
D e f i n i t i o n 2.4.1. We shall say that a bitopology (~-1,~-2) on a set X has the (i, j)-A-insertion properties, where A c 2 x is any family, if either of the following two equivalent conditions is satisfied: (1) For every subset A c X there exists a set G c A such that Ti i n t A C G c Tj clA. (2) For every pair of sets (U, F), where U c ri, F c co rj and U c F, there exists a set G c A such that U C G c F. It is obvious t h a t if (T~, r2) on X has the (i,j)-A-insertion properties, then ~, X E A. It is likewise obvious that the antidiscrete topology on X possesses the A-insertion property for any family .4 c 2 x .
2.4. Bitopological Insertions
93
R e m a r k 2.4.2. The following implications hold for a BS (X, 71 < 72) and any family A c 2x: ('rl,r2)
has the 2-A-insertion property ~ ( r l , r 2 )
has the (1,2)-A-insertion property
(rl,r2) has the (2,1)-A-insertion property ==>(rl,r2) has the 1-A-insertion property
so that (T1,T2) has not the l-A-insertion property ===~(Tl,r2) has not the (1,2)-A-insertion property (T1,T2) has not the (2,1)-A-insertion property ==*(~-l,T2) has not the 2-A-insertion property.
Important information on the (i, j)-A-insertion properties is given by the next obvious statement. P r o p o s i t i o n 2.4.3. The following conditions are satisfied for a BS (X, T1, 7-2)2 (1) I r A c 2 x is any family, then (71,72) has the (i,j)-A-insertion properties if and only if (71,72) has the (j, i)-coA-insertion properties. (2) If A1 c A2 c 2 x are any families and (71,72) has the (i,j)-Al-insertion properties, then (71,72) has the (i,j)-A2-insertion properties. (3) If A c 2 X is any family and (71,72) has the (i,j)-A-insertion properties, then (71,72) has the (i,j)-Ad-insertion properties, where Ad c {i-Int A, j-C1 A, (i, j)-Int C1 A, (j, i)-C11nt A }. R e m a r k 2.4.4. Naturally, much importance is attached to the nontrivial aspects of the (i, j)-A-insertion properties, in particular, to cases where a bitopology (rl, r2) on X has the ( i , j ) - ( A \ A4)-insertion properties or (i,j)-B-insertion properties, where B c A4 c A and the bitopology (71,72) on X always has the (i, j)-A4-insertion properties. For example, the nontrivial Ga(X)-insertion property, that is, the (Ga(X) \ r)-insertion property (the insertion of ~a-sets method in the author's terms) was used in [1731 to prove that certain functions belong to the Baire class one. It is clear that there are also nontrivial aspects of bitopological (and hence topological) insertions that are distinct from those mentioned above, when, in particular, the distinctions are conditioned only by the fact that topologies 71 and 72 on X are not comparable. Some of these aspects can be helpful in characterizing the r and N-relations. Theorem
2.4.5. The conditions below are equivalent for a BS (X, 71,72):
(I) 7-i is coupled to ~2. (2) (T1,7"2) has the (2, 1)-l-OD(X)-insertion property (,~-->, (T1,7-2) has the (1, 2)-l-CZ)(X)-insertion property). (3) (71,72) has the (2,1)-rl-insertion property (<---->, (71,72) has the (1, 2)-co rl-insertion property). (4) For every set U E r2 there exists a set V E 7-1 such that U c V and T1 c 1 U :
T1 c 1 V .
Proof. Clearly, by virtue of (1) of Proposition 2.4.3 for the equivalences in (2) and (3) it suffices to consider only the corresponding left sides.
94
II. Different R e l a t i o n s Between T w o Topologies . . .
(1) = ~ (2). Let 7-1 be coupled to 7-2 and A C X be any subset. T h e n by (2) of T h e o r e m 2.2.6, r2 int 7-1 el A C rl int 7-1 C1A and if V - 7-1 int 7-1 el A, then 7-2 int A c V c 7-1 clA. It is evident t h a t V E I - O D ( X ) and by (1) of Definition 2.4.1, (7-1,7-2) has the (2, 1)-l-OZ)(X)-insertion property. On the other hand, by (2) of Proposition 2.4.3, the inclusion 1 - O D ( X ) C 7-1 gives t h a t ( 2 ) ~ (3). (3) ~ (1). If A c X is any subset, then there exists a set V E 7-1 such t h a t 7-2 int A c V c T 1 el A. Hence 7-2 int A c 7-1 int 7-1 c1A and by (3) of T h e o r e m 2.2.6, 7-1 is coupled to 7-2. (3) ----5, (4). If U E 7-2 is any set, then by (2) of Definition 2.4.1 there exists a set V E 7-1 such t h a t U c V c 7-1 cl U. It is obvious t h a t 7-1 cl V - T 1 cl U. (4) ~ (3). Let U c 7-2, F c co 7-1 and U c F . Then there exists a set V c 7-1 such t h a t U C V C 7-1 el g C F and so U C V C F. Hence it remains to use (2) of Definition 2.4.1. D C o r o l l a r y 2.4.6. The following conditions are equivalent for a BS (X, 7-1,7-2)" (1)
T1
(2)
T1
7"2"
and T 2 are weakly connected in the sense of [182].
Proof. The proof follows directly from Definition 1 in [182] and (4) of Theorem 2.4.5. D Theorem
2.4.7. The following conditions are equivalent for a BS (X, 7-1,7-2):
(1) 7-1 i8 near 7-2.
(2) (7-1,7-2) has the 2-(1,2)-OZ)(X)-insertion property (z---->. (7-1,T2) has the 2-(1, 2)-CZ)(X)-insertion property). (3) (7-1,7-2) has the 2-7-1-insertion property (r (7-1,7-2) has the 2-coT-l-insertion property). (4) For every set U E 7-2 there exists a set V E 7-1 such that U C V and 7-2 cl U = 7-2 cl V.
Proof. We consider only the left sides of the equivalences in (2) and (3). (1) ==~ (2). Let T 1 be near r2 and A c X be any subset. Then by (3) of Theorem 2.a.7, r2 int A c 7-1 int 7-2 cl A and if V = 7-1 int 7-2 cl A, then V E (1, 2 ) - O D ( X ) , 7-2 i n t A c V c 7-2clA and so (7-1,7-2) has the 2-(1,2)-OD(X)-insertion property. F u r t h e r m o r e since (1, 2 ) - O D ( X ) c rl, by (2) of Proposition 2.4.3, (2) ~ (a). (3) ~ (1). If A c X is any subset, then there exists a set V ~ 7-1 such t h a t 7-2 int A c V c 7-2 clA. Hence 7-2 int A c 7-1 int 7-2 clA and by (3) of T h e o r e m 2.3.7, 7-1 is near 7-2. (3) ----5, (4). If U E 7-2 is any set, then by (2) of Definition 2.4.1 for i = j = 2 there exists a set V c 7-1 such t h a t U C V C 7-2 el U. Clearly, 7-2 cl U = 7-2 cl V. (4) ~ (3). Let U ~ r2, F E co r2 and U c F. Then there exists a set V E rl such t h a t U c V and 7-2 el U = 7-2 el V so t h a t U c V C F and thus (3) is satisfied. 89 We conclude the discussion of this section by introducing some new notions of bitopological insertions which will be used in C h a p t e r III to describe small and large inductive (i, j ) - d i m e n s i o n functions.
2.4. Bitopological Insertions
Definition
2.4.8.
95
We shall say that a bitopology (7-1,7-2) on X has the
(i,j)-(M,A)-insertion properties, where A4, A c 2 x are any families, if either of the following two equivalent conditions is satisfied: (1) For every set A c Ad and its a n y / - o p e n neighbohood U(A) there exists a set G c A such that U(A) c G c rj cl U(A). (2) For every set A E M and any sets U E 7-i, F E co rj together with A c U c F there exists a set G c A such that U c G c F. R e m a r k 2.4.9. It is clear that if (7-1,7-2) Oil X has the (i, j ) - ( M , A)-insertion properties, then X E A. Also 2~ c Ad implies ~ c A, but in contrast to Definition 2.4.1 it is not necessary t h a t the condition ~ c A be fulfilled. The following implications are correct for any families A/I, A of subsets of a BS (X, T 1 < 7-2):
(rl,r2) has the 2-(2M,.A)-insertion property
==> (rl,r2) has the (1,2)-(2M,A)-insertion property
(rl,r2) has the (2,1)-(2td,A)-insertion property ==> (rl,r2) has the 1-(M,A)-insertion property. P r o p o s i t i o n 2.4.10. Let A , M and 13 be any families of subsets of a BS (X, 7-1,7-2) and ;g c M . Then the following three conditions are equivalent: (1) (T1,7.2) has the
(i,j)-A-insertion properties.
(2) (7-1,7-2) has the (i,j)-(2X,A)-insertion properties. (3) (7-1,7-2) has the (i,j)-(M,A)-insertion properties.
Moreover, each of the conditions (1)-(3) implies (4) (7-1,7-2) has the (i, j ) - ( ~ , j4)-insGrtioft properties.
Proof. It is obvious that (1) = ~ (2) ==~ (3). On the other hand since ~ c M and 2~ c A for every subset A c X so that ~ c U c F, where U c 7-~, F c co7-j, (2) of Definition 2.4.8 implies (2) of Definition 2.4.1 and so (3) = ~ (1). The implication (2) ---> (4) is also obvious, but the inverse implication does not hold. D Hence it is clear that if 2~ c ,4, then (7-1,7-2)on X has the (i,j)-A-insertion properties e---->, (7-1,7-2) on X has the (i, j)-(A, A)-insertion properties. E x a m p l e 2.4.11. Let X = { a, b, c, d, e, f }, 7- = { Z , { a , b , c } , { a , b , c , d } , X } , A = {{a, b, c, d, e}, X}, and M = {{a}, {a, b}}. Then 2~gA/I, 7- on X has the ( M , A)-insertion property and does not have the A-insertion property so t h a t 7does not have the (2x, A)-insertion property since for the set {f} together with int{f} = ~ and cl{f} = { e , f } , there does not exist a set G E A such t h a t G C {e, f}. C o r o l l a r y 2.4.12.
The following conditions are satisfied for a BS (X, 7-1,7-2):
(1) If M and A are any families of subsets of X and ;g c M , then (7-1,7-2) has the (i,j)-A-insertion properties ,e-> (7-1,7-2) has the (i,j)-coA-insertion properties e----->, (7-1,7-2) has the (i, j ) - ( M , A)-insertion properties.
96
II. D i f f e r e n t R e l a t i o n s
Between Two Topologies ...
(2) If A1,A2 and M are any families of subsets of X, where .41 c A2 and (7-1,72) has the (i,j)-(A/t, A1)-insertion properties, then (7-1,7-2) has the (i, j)-(Ad, A2 )-insertion properties. (3) If M I,A/I2 and A are any families of subsets of X, where .All C .Ad2 and (T1,T2) has the (i,j)-(A/12, A)-insertion properties, then (T1,T2) has the (i, j ) - ( M 1, A)-insertion properties.
Pro@ The proof follows from Definition 2.4.8, Propositions 2.4.3 and 2.4.10.
D
C H A P T E R III
Dimension of B itopological Spaces The notion of a zero-dimensional BS was introduced by I. L. Reilly [216] on the basis of the idea of bitopological disconnectedness as examined by J. Swart [248]. A systematic study of bitopological dimension functions was undertaken independent of one another by M. Jelid [144], [145]; D. (~irid [66], and us [84], [86], [87], [101], [102]. As distinct from [66], [144], [145], the ideas set forth in [84], [86], [87], [101], [102] were essentially based on the notion of bitopological boundaries. The nine functions corresponding to the small inductive dimension [107], [179], [254], the large inductive dimension [45], [82], and the covering dimension [168], [192], [11], [111] of a TS are defined for each integer n > - 1 . Each definition is followed by stating and proving the respective properties of these functions. By analogy with [11] the p-small and p-large inductive dimensions are formulated in terms of both bitopological partitions and neighborhoods in a manner such that for n - 0 p-small inductive dimension leads to the notion of Reilly. In the aboveindicated succession of dimension functions, monotonicity with respect to arbitrary BsS's is proved for the first three functions, while monotonicity with respect to the p-closed subsets is established for the remaining six functions. Further, interrelations of the p-inductive dimensions and their topological versions are considered when topologies are comparable by inclusion or are coupled,
Using different methods, we give two equivalent definitions of the p-small inductive dimension p - i n d X for every nonnegative integer n. D e f i n i t i o n 3.1.1. Let (x,A) be a pair in a BS (X, T1,7-2) such that A c coTi and x g A. Then we say that a p-closed set T is a partition, corresponding to the
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3.1. Pairwise Small Inductive Dimension
99
Proof. (1) Let ( i , j ) - i n d X - n < oc. I f x c X, F c cori a n d x - g F , then by ( 2 ) o f Definition 3.1.3, there exists a partition T, that is, X \ T - HI U H2, H~ c Ti \ {2~}, x c Hi, A c Hj and Hi N Hj - ;g. Therefore (X, rl, r2) is (i,j)-regular; for the rest it remains to use Corollary 2.2.9 and Theorem 1 in [258], respectively. (2) It sumces to prove that ( i , j ) - i n d X - k implies ( i , j ) - i n d Y < k. For k - - 1 , k - co, the statement is correct. Let it be correct for k_< n - 1 . We shall prove the statement for k - n. If x r Y, A' c co r'i, and x ~ A', then there exists a set A c co ri such that A' - A n Y. Since (i, j ) - i n d X - n, to the pair (x, A) there corresponds a partition T such that (i, j ) - i n d T <_ n - 1. It is obvious that T ' - T n Y is the partition, corresponding to ( x , A ' ) i n (Y, r{, r~). Hence, by induction, (i, j ) - i n d T' _< n - 1 since T' C T. D C o r o l l a r y 3.1.5. The statements below hold for a BS (X, rl, r2)" (1) /f p - i n d X is finite, then (X, rl, r2) is p-regular. (2) / f ( K r { , r ~ ) i s any BsS of (X, r~, r,e), then p - ind Y < _ p - i n d X . C o r o l l a r y 3.1.6. Let (X, rl, r2) be a BS and n denote a nonnegative integer. Then:
(1) ( i , j ) - i n d X <_ n if and only if for every point x c X and any i-neighborhood U(x) there exists an i-open neighborhood V ( x ) such that rj cl V ( x ) c U(x) and ( i , j ) - i n d ( j , i ) - F r V ( x ) < n - 1 or, equivalently, (X, r1,72) has an i-base gi such that (i, j ) - i n d ( j , i ) - F r V _< n - 1 for every V c gi. (2) If (X, 7-1,7-2) i8 i-second countable, then ( i , j ) - i n d X < n if and only if (X, rl, r2) has a countable i-base Bi such that (i, j ) - i n d ( j , i)- Fr V _< n - 1 for every V c 13i. Proof. It suffices to prove only the first equivalence in (1). Let (i, j ) - i n d X <_ n, x c X and U(x) be any /-neighborhood. It can be assumed without loss of generality that U(x) c 7i. Then for the pair ( x , A - X \ U(x)), there exists a partition T such that X \
-
u H2,
and (i, j ) - i n d T < n -
, A c Hh,
xcH
H t n H2 -
1. It is evident that
rj cl Hi CqHj - ~ and so rj cl Hi c X \ A - U(x). Let Hi - V(x). Then x c
c
cl
c
and by (2) of Remark 3.1.2 and (2) of Proposition 3.1.4, (i, j ) - i n d ( j , i)- Fr V ( x ) < n - 1. Conversely, let the condition in the right-hand part of the equivalence be satisfied and (x, A), where A c co 7i, x c A, be any pair. Then U(x) - X \ A is an /-open neighborhood of x and, by condition, there exists a n / - o p e n neighborhood V ( x ) such that rj c l V ( x ) C U(x) and ( i , j ) - i n d ( j , i ) - F r V ( x ) < n -
1.
100
III. Dimension of Bitopological Spaces
But following (1) of Remark 3.1.2, (j, i)-Fr V ( x ) i s the partition corresponding to (x, A) and thus it remains to apply (2) of Definition 3.1.3. D Hence by (1) of Corollary 3.1.6, we have p - i n d X - 0 ,z--v, (7-1 has a base consisting of 2-closed sets and ~-2 has a base consisting of 1-closed sets) [216]. C o r o l l a r y 3.1.7. The Cartesian product O<3
O<3
(X3
t=l
t=l
t=l
of a ~o~tabl~ fa.~ly {(X.. ~. ~)}7=1 of BS'~ ~ ( ~ . j ) - ~ o d ~ n ~ o n a l (a~d. of ~ o ~ . p-z~o d~.~~o~al) if a~d o~ly if all BS'~ (X,. ~{. ~) ~ (~.j)-z~o d~mensional (p-zero dimensional). Proof. If X # ;~, then each BS (Xt, 7-~,7-~) is d-homeomorphic to a BsS of (X, T1, ~-2) so that if (X, 71,7-2) is (i, j)-zero dimensional, then by (2) of Proposition 3.1.4 and Remark 3.3.7 (following which (i, j ) - i n d X are bitopological properties), all BS's (Xt, T{, ~-~) are (i, j)-zero dimensional. To prove the reverse implication, it is sufficient to consider, for each t - 1, oc, a n / - b a s e B~ for the BS (Xt, 7-~, 7-~), consisting o f / - o p e n and j-closed sets and to recall that sets of the form O<3
k=mq-1
where U~ c B~ for t < m and m - 1, oc, constitute an /-base for (X, rl, 7-2) and a r e / - o p e n and j-closed in (X, 7-1,7-2)" [-7 Proposition
3.1.8. I f f o r a BS (X,~,r2), we have ( i , j ) - i n d X - n, n >_ 1, 1, the BS (X, 7-1,7-2) contains a p-closed subset Y such
then for each k - O, n that (i, j)- ind Y - k.
Pro@ It is enough to show that X contains a p-closed subset Y such that ( i , j ) - i n d Y - n - 1 since it is not difficult to see that p-Cl(Y) c p-Cl(X) for every set Y c p-Cl(X). Since (i, j ) - i n d X > n - 1, there exist a point x c X and a n / - n e i g h b o r h o o d U(x) such that for e v e r y / - o p e n neighborhood V(x), satisfying the condition Tj cl V(x) c U(x), we have (i, j ) - i n d ( j , i)- Fr V(x) > n - 2. On the other hand since ( i , j ) - i n d X that
<_ n, among V(x) there exists V'(x) such
7-jclV'(x) c U(x) and ( i , j ) - i n d ( j , i ) - F r V ' ( x ) <_ n Therefore for the p-closed subset Y (i, j ) - i n d Y - n - 1.
1.
(j, i)- Fr V'(x) of (X, 7-1,7-2), we have K]
C o r o l l a r y 3.1.9. If for a BS (X, 7-1 < 7-9.), we have ( i , j ) - i n d X - n, n >_ 1, 1, the BS (X, T1 < ~-2) contains a 2-closed subset Y such
then for each k - O, n that (i, j)- ind Y - k.
3.1. Pairwise Small Inductive Dimension
Proposition
3.1.10.
101
The conditions below are satisfied for a BS
(X, 7"1,7"2):
(1) If (X, 7"1,7"2) is (i,j)-regular and p-extremally disconnected, then (i,j)-indX =0. (2) If p - i n d X = 0, then (X, 7"1,7"2) is p-completely regular and, thus, it is
also p-almost completely regular. Proof. (1) Let x c X and U(x) be any /-neighborhood. T h e n by (2) of Proposition 0.1.7, there exists an /-open neighborhood V(x) such t h a t 7"j c l V ( x ) C U(x). But by (3) of Definition 0.1.18, 7"j cl V(x) is a l s o / - o p e n . Therefore W(x) = 7"j cl V(x) r 7"i O co T"j and, following (9) of T h e o r e m 1.3.2, (j, i ) - F r W(x) = ~ so t h a t (i, j) ind X = 0. (2) Let p - i n d X = 0. T h e n 7"1 has a base consisting of 2-closed sets and 7"2 has a base consisting of 1-closed sets. If Zi = {A : A c co 7"i N 7"j}, then it is not diMcult to see t h a t Z = {Z1, Z2} is a p-normal base of the BS (X, T~, T2) and, by T h e o r e m 0.2.5, (X, 7"1,7"2) is p-completely regular. The rest follows from (13) of Definition 0.1.6. [--1
7"1,7"2) is
C o r o l l a r y 3 . 1 . 1 1 . If (X, BS, then p - ind X - 0.
a p-regular and p-extremally disconnected
Note also here t h a t a BS (X, 7"1,7"2) is called hereditarily p-disconnected if X does not contain p-connected subsets of cardinality larger t h a n one. Proposition 3.1.12. Every R-p-T1 p- ind X - 0, is hereditarily p-disconnected.
(i.e., d-T~) BS (X, 7"1,7"2), where
Pro@ Let A C X be any subset and ]A I > 1. If Xl, X2 ~ A, Xl ~ X2, then p- i n d X - 0 implies t h a t , there is a set U c 7"2 Nco7.1 such t h a t Xl ~ U c X \ {x2} since X \ {x2} c 7"1 n CO7"2. Therefore !
I
A O U e (~-2 N co 7"1) \ { •}
i
l
and A \ U e (7-1 n CO "/-2)\ { e }
for the BsS (A, 7"~, 7"~) so t h a t A is not p-connected.
Q
R e m a r k 3 . 1 . 1 3 . If for a given fixed point x E X and any pair (x, A), where A c co7"i, x-~A, there exists a partition T such t h a t ( i , j ) - i n d T < n - 1, t h e n we write (i, j)- indx X < n. The meaning of (i, j)- indx X - n or (i, j)- indx X - oc is clear and, we have ( i , j ) - i n d X - s u p { ( i , j ) - i n d x X " x c X } . E x a m p l e 3 . 1 . 1 4 . Let (X, 7"1) be a TS and B c X be a fixed proper subset of X. It is clear t h a t if 7"2 is the B - t o p o l o g y on X (i.e, 7"2 - 7"(B) - {X} U {A c X " n E 2 B} [14, p. 63]) and if (X, 7"1,72) is l-T1, then (2, 1)-indx X - 0 for each point x c B. L e m m a 3 . 1 . 1 5 . Let x be a fixed point of a subset Y c X , where (X, 7"1,7-2) is a hereditarily p-normal BS. Then (i,j)-indx Y < n if and only if for every i-neighborhood U(x), there exists an i-open neighborhood V(x) such that
V(x) c U(x) and (i, j)-ind (Y N (j, i ) - F r V(x)) < n - 1.
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3.1. Pairwise Small Inductive Dimension
103
Conversely, assume t h a t the condition of L e m m a 3.1.15 is satisfied. We shall show t h a t (i, j)-indx Y <_ n. Since a n y / - n e i g h b o r h o o d U(x) contains a n / - o p e n neighborhood V(x) such t h a t Tj cl V(x) c U(x), it is clear t h a t a n y / - n e i g h b o r h o o d U'(x) - U(x) N Y contains a neighborhood V'(x) - V(x) N Y C 7-'i, and
cl
c
cl
n Y, z n ( X \ V ( x ) ) - Z \
so t h a t (j, i)- Fr~ V'(x) c Y a (j, i)- Fr V(x). Hence, by monotonicity,
( i , j ) - i n d ( j , i ) - F r , V'(x) < ( i , j ) - i n d ( Y N ( j , i ) - F r V ( x ) ) < n -
1.
Thus (i, j)- ind~ Y < n.
V~
C o r o l l a r y 3 . 1 . 1 6 . Let ~ r Y c X , where a BS (X, T1,T2) is hereditarily p-normal. Then ( i , j ) - i n d Y = 0 if and only if for each point x c Y and every i-neighborhood U(x), there exists an i-neighborhood V(x) such that V(x) c U(x) and Y O (j, i ) - F r V(x) = ~. Moreover, if (X, ~-1,T2) is i-second countable, then ( i , j ) - i n d Y = 0 if and only if (X, T1,Tv) has a countable i-base Bi such that Y N (j, i ) - F r V = 2~ for every V c Bi.
Proof. The first part is obvious. The second part follows directly from the first part in conjunction with (2) of Corollary 3.1.6. 89 Theorem 3.1.17 (A Generalized F o r m u l a ) . The inequality
Version
of the
Menger-Urysohn
( i , j ) - i n d ( M U N) < ( i , j ) - i n d M + ( i , j ) - i n d N + 1 holds for any subsets M and N of a hereditarily p-normal BS (X, T1,7-2). Proof. If (i, j ) - i n d M - oc or (i, j)- ind N - oc, then the inequality is obvious. It can therefore be assumed t h a t (i, j ) - i n d M - m, (i, j)- ind N - n, where - 1 < m < oc, - 1 <_ n < oc, are the whole numbers. We shall prove by an inductive a s s u m p t i o n with respect to the n u m b e r m + n t h a t ( i , j ) - i n d ( M U N) <_ m + n + 1.
(,)
Inequality (,) holds for m + n - - 2 . Let us assume t h a t (,) holds for m + n < k - 1 and prove it for m + n - k. Let x c M U N be any point and U(x) be any /-neighborhood from ( M U N, r~, r~). It can be assumed w i t h o u t loss of generality t h a t x c M. Since (i, j ) - i n d M - m, we have (i, j)-ind~ M <_ m and, following L e m m a 3.1.15, there exists a neighborhood V(x) c ri such t h a t
V(x) C U(x) and (i, j ) - i n d ( M N (j, i)-FrA,uN V(x)) <_ m On the other hand, (i, j ) - i n d ( N N (j, i)- FrMu N V(x)) < (i, j ) - i n d N < n. Let M 1 - ~ : / n (j, i)- FI'~luN V ( x ) ,
N1 -- N N (j, i)- FraiuN V(x).
Since
(i,j)-indM1 <_m-l,
( i , j ) - i n d N l <_ n
and
( i , j ) - i n d M 1 + ( i , j ) - i n d N 1 <_ m -
1 + n <_ k - 1,
1.
104
III. Dimension of Bitopological Spaces
from inequality (,) it follows that (i, j)-ind(M1 U N1) _< (m - 1) + n + 1 - m + n. But M1 U N1 - ( M N (j, i)-FrMu N V ( x ) ) U ( N N (j, i)-FrMu N V(x)) -= (j,
and so (i, j)- ind(j, i)- FrM.~ V ( x ) <_ m + n. Since V ( x ) is any small/-open neighborhood of an arbitrary point x c M U N (i.e., V ( x ) is contained in any given /-neighborhood U(x)), we ascertain that (i, j ) - i n d ( M U N) _< m + n + 1. C o r o l l a r y 3.1.18. The following statements hold for a hereditarily p-normal BS (X, 7-1,7-2)" (1) If M and N are any subsets of X , then
p - i n d ( M U N) _< p - i n d M + p - i n d N
+ 1.
(2) If Mo, M 1 , . . . ,M~ are any subsets of X , then (i, j ) - i n d (M0 U M1 U . . . U M~) _< <_ ( i , j ) - i n d M o + ( i , j ) - i n d M 1 + . . . + ( i , j ) - i n d M n + n and thus
p - i n d (M0 U M 1 U . . . U Mn) < p - i n d M0 + p - i n d M1 + . . . + p - i n d Mn + n. n
(3) If X
-
U Mk, where ( i , j ) - i n d M k
-
0 for each k -
O, n, then
k--0 n
(i, j)- ind X <_ n and thus, we have p - i n d X <_ n for X -
U Mk, where k=0
p - ind Mk - - 0 for each k -
0, n.
T h e o r e m 3.1.19. If a R-p-T1, d-second countable, and p-normal BS (X, 7-1,7-2) can be represented as a union of two BsS's Y and Z such that ( i , j ) - i n d Y < n and ( i , j ) - i n d Z < O, then ( i , j ) - i n d X < n + 1. Proof. Let x c X be any point and U(x) c 7-i be its any neighborhood. Then by Theorem 3.2.15 below, there exist disjoint sets U E 7-i and V E 7-j such that x E U, X \ U(x) c V and (X \ (U U V ) ) N Z - ;~. Clearly, x c U c V(x). But
(j, i)- Fr U-7-j cl U n (X \ U) C (X \ V) N (X \ U ) - X
\ (V U U) c X \ Z c Y
and by (2) of Proposition 3.1.4, (i, j)- ind(j, i)- Fr U _< n. n+l.
Thus (i, j)- ind X _< K]
C o r o l l a r y 3.1.20. If a R-p-T1, d-second countable, and p-normal BS (X, 7-1,7-2) can be represented as a union of two BsS's Y and Z such that p - ind Y _< n and p - ind Z <_ O, then p - ind X _< n + 1. T h e o r e m 3.1.21. If a 1-T1, d-second countable, and p-normal BS (X, 7-1 < 7-2) can be represented as a union of two BsS's Y and Z, where p - i n d Y p - ind Z - 0 and one of them is l-closed, then p - ind X - 0.
3.1. Pairwise Small Inductive Dimension
105
Proof. Let, for example, Y E COT1 C p - e l ( X ) . S i n c e X \ Y c Z, by ( 2 ) o f Corollary 3.1.5, p - i n d ( X \ Y) = 0, where X \ Y c ~1. By Corollary 0.1.13 the BS (X, T1 < T2) is p-perfectly normal and hence X \ Y E 2-~co(X), t h a t is, (X5
X \ Y -
[_J Fk, where Fk C co ~-2 C p-Cl(X) and p- ind Fk - 0 for every k - 1, oc. k=l
On the other hand, Y c p - e l ( X ) and, therefore, oo
x-ru(x\r)-ru
[_J /~=1
Thus it remains to use Corollary 3.2.28 below since (X, T1 < ~-2) is l-T1 implies ( X , T1 < 7-2) is R-p-T1. K] It is not difficult to see t h a t Theorem 3.1.21 remains valid if one of the sets Y and Z is l-open. T h e o r e m 3.1.22. If a R-p-T1, d-second countable, and p-normal BS (X, T1, T2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that (i, j ) - i n d Fk < n for each k - 1, oc, then (i, j)- ind X < n.
Proof. We shall apply induction to the number n. For n - 0 the theorem is already proved in Corollary 3.2.28 below. Assume that the theorem holds for n 1 and oo
prove it for n. Let X -
[.J Fk, where Fk
-
T1 el f k n 7-2 el f k
and (i, j)- ind Fk <_ n
k=l
for each k - 1, oc. By (2) of Corollary 3.1.6 choose for k - 1, oc a countable/-base B) for the BsS Fk such t h a t (i, j ) - i n d ( j , i)-Fr U < n - 1 for every U c B~. By the inductive assumption, for the BsS (N2)
Y-
{U(j,i)-FrU'Uc
UB~} k=l
of the BS (X, T1,72), we have the inequality (i, j)- ind Y <_ n - 1. Now, by the second part of Corollary 3.1.16, for each k - 1, oc, the BsS Zk - Fk \ Y of a BS (Fk, ~-~,~-]) has ( i , j ) - i n d Z k <_ O. Therefore, by Corollary 3.2.28 below, the BsS oo
Z -
U zk - x \ Y of ( x , 7-1, ~-2) satisfies the inequality (i, j ) - i n d Z <_ 0 because k=l
from the equalities
Z k - F k \ Y - - F k N Z - ( T 1 clFk N 72 clFk) N Z--(T1 clFk N Z ) N (72 clFk N Z) it follows t h a t all the Zk's are p-closed in Z. Thus, by virtue of Theorem 3.1.19, we have (i, j ) - i n d X <_ n. [-1 C o r o l l a r y 3.1.23. If a R-p-T1, d-second countable, and p-normal BS (X, T1,7-2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that p - ind Fk < n for each k - 1, oc, then p - ind X < n. C o r o l l a r y 3.1.24. If a 1-T1, d-second countable, and p-normalBS (X, T 1 < 7-2) can be represented as a union of a sequence F1,F2,... of i-closed sets, where p - ind Fk < n for each k - 1, oc, then p - ind X < n.
Pro@ The proof follows directly from the inclusions co ~-1 c co ~-2 C p-Cl(X).
E]
106
III. Dimension of Bitopological Spaces
Corollary 3.1.25. If a R-p-T1, d-second countable, and p-normal BS (X, ~-1,7~) can be represented as a union of a sequence F1, F 2 , . . . , where every O0
Fk is a countable union of p-closed sets, that is, Fk -- U F ~ and p - ind Fk _< n m=l
for each k - 1, oo, then p - ind X < n. Proof. By (2) of P r o p o s i t i o n 3.1.4, p - i n d F ~ OO
< n for every k - 1, oo, m -
1, oc
CxD
U
o
k=l m=l
Corollary 3.1.26. If a l-T1, d-second countable, and p-normal BS (X, T1 < T2) can be represented as a union of two B s S ' s Y and Z, where p - ind Y _< n, p - ind Z <_n and one of them is 1-closed or 1-open, then p - ind X <_ n.
Pro@
Let Y c T1.
Then, by Corollary 0.1.13, Y c 2 - ~ ( X ) s o
that Y -
OO
U Fk, where Fk E coT2 C p - C l ( X ) for each k -
1, oc.
Since X \ Y
C Z, we
k=l
have p - i n d ( X \ Y) < n. Clearly, p - i n d Fk < n for each k p - ind X _< n because
1, oc and, therefore,
OO
X - Y U ( X \ Y ) - U Fk U ( X \ Y). k=l
T h e p r o o f for a 1-closed set is similar.
[-]
Theorem 3.1.27. For a R-p-T1, d-second countable, and p-normal BS (X, rl,r2), we have: ( i , j ) - i n d X < n if and only if X can be represented as a union of two B s S ' s Y and Z such that (i, j ) - i n d Y < n - 1 and (i, j ) - i n d Z < 0.
Proof. Let ( i , j ) - i n d X <_ n. T h e n , by (2) of Corollary 3.1.6, X has a c o u n t a b l e /-base Bi such t h a t ( i , j ) - i n d ( j , i ) - F r V < n - 1 for every U c Bi. By Theorem 3.1.22 the BsS Y - U { ( j , i ) - F r U " U e B~} has ( i , j ) - i n d Y <_ n - 1 and by the second p a r t of Corollary 3.1.16, (i, j ) - i n d Z < 0, where Z - X \ Y. To complete the proof, it suffices to apply T h e o r e m 3.1.19. [] Corollary 3.1.28. For a R-p-T1, d-second countable, and p-normal BS We have p - ind X <_ n if and only if X can be represented as the union of two B s S ' s Y and Z such that p - ind Y < n - 1 and p - ind Z < 0.
( X , 7-1, T2) ,
T h e o r e m 3 . 1 . 2 9 . Let (X, 7"1,7"2) be a R-p-Tz, d-second countable, and p-normal BS such that p - ind X < n. Then for every pair A E co ~-1, B E co ~-2 with A • B - 2~, there exists a partition T between A and B such that p - ind T < n - 1.
Proof. By T h e o r e m 3.1.27, X - Y U Z, where p - i n d Y < n - 1 and p - i n d Z < 0. Following T h e o r e m 3.2.15 below for A and B, t h e r e exists a p a r t i t i o n T such t h a t T~Z-2~. SinceTcX\ZcY, wehavep-indT_
3.1. Pairwise Small Inductive Dimension
107
Pro@ Since p- ind Y _< n, by Corollary 3.1.28, Y - P u Q , where p- ind P <_ n - 1, p - i n d Q _< 0. Let A c coT1, B c coT2 and A ~ B - 2~. Then A N Y c co~-~, B A Y ~ coT~ in (Y, ~-~,7-~) and hence, by Theorem 3.2.15 below, there exists a partition T I between A n Y and B N Y such t h a t T' C P so that p - i n d T ~ <_ n - 1. Moreover, by the second part of L e m m a 3.2.14 below, there exists a partition T between A and B such that T CqY C T'. Thus p - i n d ( T N Y) < n - 1. 9 T h e o r e m 3.1.31. For a d-second countable and p-normal BS (X, 7 1 , 7 2 ) , we have: ( i , j ) - i n d X <_ n if and only if X has a countable i-network Af~ such that (i, j ) - ind(j, i)- Ft" N _< n - 1 for each N c Aft.
Pro@ Since a n y / - b a s e is an /-network, by (2) of Corollary 3.1.6, it suffices to prove only t h a t if an i-second countable BS has a n / - n e t w o r k iV - {Nk }k~__l such that (i, j)- ind(j, i)- Fr Nk < n - 1 for every k - 1, oc, then (i, j)- ind X _< n. O(9
Let Y -
[_J (j, i)-Fr Nk and Z - X \ Y. It follows from Theorem 3.1.22 that k=l
(i, j ) - i n d Y _< n - 1. We shall show t h a t (i, j ) - i n d Z <_ 0. For an arbitrary point x E Z and a n / - n e i g h b o r h o o d U(x), there is Nk C Af~ such t h a t x c Nk C U(x). Since
x C X \ Y C X \ ( j , i ) - F r N k - X \ (rj clNk A r~ cl(X \ Nk)) --
-
rj int(X \ Nk) U ri int Nk,
we have x c Vk(x) -- ~-i int Nk C U(x). But (j, i)- Fr Vk(x) c (j, i)- Fr Nk and hence Z N (j, i)-Fr Vk(x) - 2~. Therefore, by the second part of Corollary 3.1.16, (i, j ) - i n d Z <_ 0. Thus it remains to use Theorem 3.1.19. 89 C o r o l l a r y 3.1.32. For a R-p-T1, d-second countable and p-normal BS (X, rl,T2), we have: p - i n d X <_ n if and only if X has a countable l-network A/'I and a countable 2-network A/'2 such that (1, 2)-ind(2, 1 ) - F r N < n - 1 for each
N o H 1 ~nd (2, 1)-i~d(1, 2 ) - F r M _< n - 1 for each M o H 2 Now we are going to give a new characterization of dimensions (i, j ) - i n d X in terms of the bitopological insertions introduced by Definition 2.4.8. D e f i n i t i o n 3.1.33. We say that a bitopology (T1,7"2) on X has the (i, j ) - l o c A-insertion properties if (rl, 7-2) has the (i, j)-(A//, A)-insertion properties for Ad - {{x}" x c X}. T h e o r e m 3.1.34. Let (X, T1,7"2) be an (i,j)-regular BS, let n denote a nonnegative integer, and let (rl, r2) has on X possess the ( i , j ) - l o c A - i n s e r t i o n properties, where A - { V c r~ " (i, j ) - ind(j, i)- Fr V _< n - 1}. Then (i, j ) - ind X _< n.
Pro@ Let x c X be any point and U(x) c ~-~ be its arbitrary neighborhood. By the (i,j)-regularity of (X, T1, ~-2) there exists a neighborhood V ( x ) c 7-~ such that Tj el V ( x ) c U(x). Following the condition, there is also a set V E ~-~ together with V ( x ) c V c Tj cl V ( x ) so t h a t rj cl V - Tj cl V(x). Since V is an /-open neighborhood of x and ( i , j ) - i n d ( j , i ) - F r V _< n 1 by (1) of Corollary 3.1.6, we obtain (i, j)- ind X < n. D
108
III. Dimension of Bitopological Spaces
C o r o l l a r y 3 . 1 . 3 5 . Let (X, 7-1,7-2) ((X, T)) be a p-regular BS and (7-1,7-2) has on X the (1, 2)-1oc(71 ~ co T2)-insertion and (2, 1)-lot(T2 n co n )-insertion properties. Then p - ind X - 0. It should be noted here that by Example 1 from [216], p - i n d X is, in general, different from 1-ind X and 2-ind X. Hence for a BS (X, T1, ~-2) it is natural to establish an interrelation between the dimension functions (i, j ) - i n d X and i-ind X. Recall that a point x0 of a BS (X, rl, 72) is a point of tangency of the topologies T1 and ~-2 if T1 and T2 coincide at x0, that is, if for each 1-open neighborhood U(xo) and each 2-open neighborhood V(xo), there exist a 2-open neighborhood V'(xo) and a 1-open neighborhood U'(xo) such that V'(xo) c U(xo) and U'(xo) c
V(xo) [as] 3.1.36. The conditions below are satisfied for a BS (X, T1, T2)" (1) If T1 c 7-2, then
Theorem
(1,2)-indX<
1-indX
and 2 - i n d X <
(2,1)-indX.
(2) If TICT2, then 1- ind X < (1, 2)- ind X. (3) IfTINT2, then ( 2 , 1 ) - i n d X _< 2 - i n d X . (4) If rl c 72, then x o - c n l (A) for each A E 2 x if and only if Xo is a point of tangency of 71 and T2; moreover, in this case 2-indz(, X <_ (1, 2)-indxo X <_ 1-indx(, X
and 2-indx(, X _< (2, 1)-indxo X _< 1-indx(, X.
(5) If T1
and 2 - i n d X < ( 2 , 1 ) - i n d X .
(6) / f T 1 < N T2, then 1-indX -(1,2)-indX
and 2 - i n d X - ( 2 , 1 ) - i n d X .
Proof. (1) It is clear that the inequality (1, 2)- ind X _< 1- ind X holds for 1- ind X = ec. Thus, assuming that 1- ind X - k < oc, we shall show that (1, 2)- ind X _< k. For k - - 1 the required inequality is obvious. Let us assume that this inequality is also correct for k _< n - 1 and prove it for k - n. Since 1 - i n d X - n, for every point x c X and its any 1-open neighborhood U(x), there exists a 1-open neighborhood V ( z ) s u c h that T1 clV(x) c U(x) and 1 - i n d ( 1 - F r V ( x ) ) _< n - 1. Clearly, T1 C T2 implies that r 2 cl V(x) C T1 cl V(x) and so (2, 1)- Fr V(x) c 1-Fr V(x). Hence, by the monotonicity of the small 1-inductive dimension function, we find that 1-ind(2, 1)-Fr V(x) <_ n - 1 and by the inductive assumption (1,2)-ind(2, 1)-Fr V(x) _< n - 1. Thus for every point x c X and its any i-open neighborhood U(x), there exists a 1-open neighborhood V(x) such that T2 cl V(x) C U(x) and (1, 2)-ind(2, 1)- Fr V(x) <_ n - 1. Hence (1, 2)- ind X _< n. Furthermore, by analogy with the above reasoning we prove that 2-ind X _< (2, 1)-ind X. Let us assume that the inequality holds for (2, 1)-ind X - k _< n - 1 and prove it for k - n. Since (2, 1 ) - i n d X - n, for every point x E X and its
3.1. Pairwise Small Inductive Dimension
109
any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-1 cl V(x) C U(x) a n d (2, 1)-ind(1, 2)- Fr V(x) _< n - 1. But 7-1 C 7-2 implies that 7-2 cl V(x) c 7-1cl V(x) so that 2- Fr V(x) c (1, 2)- Fr V(x). Hence, by the monotonicity of the small (2, 1)-inductive dimension function, we find that (2, 1)-ind(2- Fr V(x)) <_ n - 1 and by the inductive assumption 2-ind(2- Fr V(x)) <_ n - 1. Thus for every point x E X and its any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-2 cl V(x) C U(x) and 2-ind (2-Fr V(x)) < n - 1. Therefore 2-ind X < n. (2) By analogy with (1) let us suppose that the required inequality is correct for ( 1 , 2 ) - i n d X - k <_ n - 1 and prove it for k - n . If ( 1 , 2 ) - i n d X - n, then for every point x c X and its any 1-open neighborhood U(x), there exists a 1-open neighborhood V(x) such that 7-2 cl V(x) c U(x) and (1, 2)-ind(2, 1)- Fr V(x) <_ n - 1. But by Definition 2.2.1, 7-1 cl V(x) C 7-2cl V(x) and, therefore, 1-Fr V(x) C (2, 1)- Fr V(x). Hence, by the monotonicity of the small (1, 2)-inductive dimension function, we have (1, 2)-ind(1-Fr V(x) _< n - 1 and by the inductive assumption 1-ind(1-FrV(x) _< n - 1. Thus for every point x C X and its any 1-open neighborhood U(x), there exists a I-open neighborhood V(x) such that T1 C1 V(x)
C U(x)
and
1-ind
(1-Fr V(x)) < n - 1.
Therefore 1 - i n d X < n. (3) Let us assume that the required inequality holds for 2-ind X - k _
1.
Thus for every point x c X and its any 2-open neighborhood U(x), there exists a 2-open neighborhood V(x) such that 7-, cl V(x) C U(x) and (2, 1)-ind(1, 2)- Fr V(x) < n - 1. Therefore (2, 1)- ind X <_ n. (4) First let us prove the equivalence. If x0 c X is a tangency point of T1 and T2, then x0 c T1 clA implies x0 c T2 clA for each A c 2 x so that x0 c nl(A) for each A E 2 X. Conversely, let x 0 c n l ( A ) for each A E 2 X. If x0 is not a tangency point of 7-1 and 7-2, then there is a neighborhood U(xo) c 7-2 such that for each neighborhood V(xo) c 7-1, we have V(xo)A (X \ U(xo)) r 2~. Therefore X0 E 7 - 1 c l ( X \
U(xo) ).
gilt
Xo-~(X \ g(xo) )
-
7-2cl(X \
g(xo)))
and
so x 0 E
n l ( X \ U(xo)), which is impossible. Now, let Xo E X be a tangency point of 7-1 and 7-2. Since 7-1 C 7-2, by (1) above, it suffices to prove only that 2- indx,, X _< (1, 2)- indxl, X and (2, 1)- indx,, X <_ 1- indx,, X.
110
III. Dimension of Bitopological Spaces
It is obvious t h a t the inequality 2 - i n d x , , X < ( 1 , 2 ) - i n d x o X holds for (1, 2)-indx,, X - oc. Thus assuming that (1, 2)-indx,, X - k < oc, we shall show that 2-indx,, X < k. For k - - 1 this inequality is obviously fulfilled. Let us suppose that it also holds for k 5 n - 1 and prove it for k - n. Let U(xo) c r2 be any neighborhood. Since x0 is a tangency point of rl and r2, there is a neighborhood W ( x o ) c rl such that W ( x o ) c V(xo) and since (1,2)-indx,, X - n, there is a neighborhood V(xo) c rl such that
r 2 c l V ( x o ) C W ( x o ) and (1, 2)- ind(2, 1 ) - F r V ( x 0 ) _< n -
1.
By (2) of Proposition 3.1.4, we have (1, 2)-ind2- Fr V(x0) _< (1, 2)-ind(2, 1)-Fr V(x0) _< n and, by induction, 2 - i n d 2 - F r V(xo) _< n V ( x o ) E T 1 C T 2 such that
1
1. Thus for each U(xo) c r2, there is
r2 cl V(xo) C U(xo) and 2- ind 2- Fr V(xo) < n - 1. Hence 2-indxo X < n, that is, 2-indx0 X < (1, 2)-indxo X. Finally, let us again suppose that the considered inequality is correct for 1- indx,, X - k < n - 1 and prove it for k - n. Let U(xo) c r2 be any neighborhood. Since x0 is a tangency point of rl and r2, there is a neighborhood W ( x o ) c rl such t h a t W ( x o ) c U(xo). But 1-indxo X - n and, therefore, there is a neighborhood V(x0) c rl such t h a t T1
cl V(xo) c W ( x o ) and 1-ind 1- Fr V(xo) <_ n - 1.
By monotonicity, we have 1-ind(1, 2)- Fr V(xo) _< 1-ind 1- Fr V(xo) <_ n - 1. On the other hand, by induction, (2, 1)-ind(1, 2)- Fr V(xo) _< n - 1. Thus for each g ( x o ) E r2, there is V(xo) E rl C T2 such that T1 C1 V ( x o )
C U(xo) and
(2, 1)-ind(1, 2)- Fr V(xo) < n - 1.
Hence (2, 1)- indxo X _< n, that is, (2, 1)- indxo X _< 1- indxo X. The rest is obvious.
D
3.2. Pairwise Large Inductive D i m e n s i o n D e f i n i t i o n 3.2.1. Let ( A , B ) be a pair of subsets of a BS (X, r l , r 2 ) such t h a t A E co rj, B E co ri and A N B = ;~. Then we say that a p-closed set T is a partition, corresponding to (A, B), if X \ T = H1 U/-/2, Hi c ri \ { 2~}, H1 O/-/2 = 2~ and A c Hi, B c Hj. R e m a r k 3.2.2. In the sequel, without loss of generality, we shall sometimes consider a p a i r ( A , B ) where A E CUrl, B c cur2 and A N B - ~. As in Remark 3.1.2, it is easy to verify that in a BS (X, rl, r2) for such a pair (A, B) the following conditions are satisfied:
3.2. Pairwise Large Inductive Dimension
111
(1) If there exists a 2-open neighborhood U(A) (1-open neighborhood U(B)) such t h a t 7-1CI U(A) C X \ B
(7-2cl U(B) c X \ A ),
then (1, 2)- Fr U(A) ((2, 1)- Fr U(B)) is the partition, corresponding to (A, B) in the sense of Definition 3.2.1. (2) If T is a partition, corresponding to (A, B) in the sense of Definition 3.2.1, then (j, i)- Fr H~ C T. D e f i n i t i o n 3.2.3. Let (X, 7-1,7-2) be a BS and n denote a nonnegative integer. We say t h a t (1) ( i , j ) - I n d X - - 1 <---> X - ~. (2) (i, j ) - I n d X < n if to every pair (A, B), where A 9 co rj, B 9 co 7-i and A N B - ~, there corresponds a partition T such t h a t (i, j)- Ind T < n - 1. (3) (i, j)- Ind X - n if (i, j)- Ind X < n and the inequality (i, j)- Ind X < n - 1 does not hold. (4) (i, j)- Ind X - ~ if the inequality (i, j)- Ind X _< n does not hold for any n. As usual, p - I n d X < n ,e---->, ((1, 2 ) - I n d X _< n A (2, 1)-Ind X <_ n). L e m m a 3 . 2 . 4 . Let T be a partition in a BS (X, 7-1,7-2) which corresponds to a pair of disjoint sets (A, B) where A 9 co 7-1, B 9 co 7-2. If Y c X is a p-closed set such that A H Y # ~ # B N Y, then the set T' - T fq Y is the partition in the
Proof. By assumption, X \ T - H l t 0 H 2 , where Hi 9 7-~ \{2~}, H I N H 2 A c H2, and B c H1. Hence T - X \ (H1 O H2). It is obvious t h a t the set T' - T N Y is the required partition since Y \ T'-
- 25,
Y \ ( X \ (H1 U H2)) - (Y \ (X \ H1)) U (Y \ (X \ H2)) (YNH1) U (YAH2)-
H~ UH~,
where L e m m a 3 . 2 . 5 . If (A', B') is a pair of disjoint sets of a p-closed BsS (Y, w~, w~) of a BS (X, Wl, r2) such that A' c co w~ and B' c co 7~, then there exists a pair of disjoint sets ( A , B ) in (X, 7-1,7-2) such that A c co7-1, B c co7-2, A N Y = A', and BNY=B'.
Proof. The fact t h a t Y is p-closed in (X, 7-1,7-2) implies Y = Wl cl Y N w2 cl Y. Let / A ' 9 CO7-1,
B
~ co
l I 7-2/ in (Y, rl,r2) and
A I
N B l --~.
T h e n there are d 9 co
" %" ) and B 9 co r~" in (r2 cl Y, 7-1'" , %' " ') 7-1" in (7-1 clY, 7-1,
such t h a t A N Y = A' and B H Y = B'. It is evident t h a t A 9 co 7-1, B 9 co 7-2 and AHB=~. 89 Corollary
3 . 2 . 6 . A p-closed BsS of a p-normal BS is also p-normal.
112
III. Dimension of Bitopological Spaces
P r o p o s i t i o n 3.2.7. The statements below hold for a BS (X, 7"1,7"2)" (1) If ( 1 , 2 ) - I n d X or (2, 1 ) - I n d X is finite, then (X, 7"1,7"2) i8 p-normal. (2) For every p-closed BsS (Y, 7"~, 7"~) of ( X , 7"1,7"2), we have
( i , j ) - I n d Y <_ ( i , j ) - I n d X . (3) /f (X, 7"1,7"2) is a j-T1 BS, then ( i , j ) - i n d X <_ ( i , j ) - I n d X .
Proof. Assertion (1) is obvious. (2) It suffices to prove that (i, j ) - I n d X = k implies (i, j ) - I n d Y _< k. This inequality is correct for k - - 1 , k - oc. Let us assume that it is correct for k <_ n - 1 and prove it for k - n. If (A', B') is a pair of disjoint sets such that A' c co 7"~ and B' c co 7"~, then by Lemma 3.2.5, there exists a pair of disjoint sets (A, B) where Acco7"j,
B c c o 7 " i and A N Y - A '
BAY-B'.
Since (i, j)- Ind X - n, there exists a partition T for (A, B) such that (i, j)- Ind T _< n - 1. But by Lemma 3.2.4, T' - r A Y is the partition in (Y, 7"~,7"~), corresponding to the pair (A', B'). Hence, by the inductive assumption, (i, j)- Ind T' <_ n - 1 since T' is p-closed in T and so (i, j ) - I n d Y <_ n. The condition (3) is clear. [3 C o r o l l a r y 3.2.8. We have p - I n d Y < p - I n d X for every p-closed BsS (Y, 7"~,7"~) of a BS (X, 7"1,7"2) and, therefore, if Y E co 7"1 U CO 7"2, then p - Ind Y < p - I n d X . Moreover, if (X, 7.1,7.2) is R-p-T1, then p - i n d X _< p - I n d X . C o r o l l a r y 3.2.9. Let (X, T1, ";-2) be a BS and n denote a nonnegative integer. Then ( i , j ) - I n d X <_ n if and only if for any j-closed set F and any i-neighborhood U(F) there exists an i-open neighborhood V ( F ) such that 7.j cl V ( F ) C U(F) and (i, j)- Ind(j, i)- Fr V ( F ) < n - 1. The proof of this corollary repeats in the main that of (1) of Corollary 3.1.6, taking into account (2) of Remark 3.2.2 and (2) of Proposition 3.2.7. C o r o l l a r y 3.2.10. If for a BS (X, 7.1,7.2), we have
(i,j)-IndX-n
(p-IndX-n),
n>_l,
then for each k - O, n - 1 the BS (X, T1,7"2) contains a p-closed BsS (Y, 7"~,7"~) such that (i, j)- Ind Y - k (p- Ind Y - k). Proof. The proof is similar to the proof of Proposition 3.1.8 with Corollary 3.2.9 taken into account. D P r o p o s i t i o n 3.2.11. The following equivalences hold for every BS (X, 7"1,7"2)" (1, 2 ) - I n d X - 0 ~
(2, 1 ) - I n d X - 0 ~
p - I n d X - 0.
Proof. Let ( 1 , 2 ) - I n d X = 0, F E COT1 be any set and U(F) be its any 2-open neighborhood. Then X \ U(F) c X \ F. Hence, there exists a set V E T 1 A CO T2 such that X \ U(F) C V c X \ F and, therefore, F C X \ V c U(F) where X \ V E 7.2 • co 7.1. Thus (2, 1)- Ind X = 0. The inverse implication can be proved in a similar manner. D
3.2. Pairwise Large Inductive Dimension
113
Hence by (3) of Proposition 3.2.7 for every R-p-T1 BS (X, 71,72), we have (1, 2 ) - i n d X -
0 ,e---->, (2, 1 ) - i n d X ,e----f, p_ ind X -
0.
Theorem 3 . 2 . 1 2 . If for a d-second countable BS (X, 71,72) the equality p - ind X - 0 holds, then for every pair of disjoint sets A c co 71 and B c co 72 the empty set is a partition between them, that is, there exists a set V c 72 N co 7-1
such that A c V, B c X \ V and so p - I n d X - O. Proof. Since p- ind X - 0, for each point x c X , there exists a set U(x) c 71Gco 72 or a set V(x) E r2 n cot1 such t h a t A n u(.)
-
e,
o,. B n v ( . )
-
It is evident t h a t for the p-open covering b/ = {{U(x)}, { V ( x ) } I x e x there is a countable subfamily b/'-
{{U(xk)"
k-l,
oo},{V(xv)"
of X,
p-l,~}}
which is also a > o p e n covering of X. T h e sets Uk = U(xk) \ U V(Xp) are 1-open p
h i " - {{Uk " k -
l, oc}, {Vp " p -
l, e c } }
is also a p-open covering of X. It is obvious t h a t UkNA-2~
for each k - l ,
oc and Vv N B - 2 ~
for each p - l ,
oc.
Let V - U{Vp" A N Vp -fl 2~}. T h e n A C V. Since B N Vp - 2~ for each p - 1, oc, (x)
we have B n ( U vp) - 2~ and, therefore, p=l (x)
Box\
Uv,,cx\U{v,,
9
G
p=l
Corollary 3.2.13. If (X, 71,72) is R-p-T1 and d-second countable, then p - I n d X - 0 <--> p - i n d X - 0.
Proof. By (3) of Proposition 3.2.7, p - I n d X - 0 ----> p - i n d X - 0. T h e inverse implication is an i m m e d i a t e consequence of T h e o r e m 3.2.12 and Definition 3.2.3. [E] L e m m a 3 . 2 . 1 4 . Let (Y, r~, r~) be a BsS of a hereditarilyp-normal BS (X,rl,r2) and A r cot1, B c cot2, with A n B = ~. Then for every partition T' in the BsS (Y, w[, T~) between Y N wl cl V1 and Y AT2 cl V2, where Pl and P 2 age 2-open and l-open subsets of X , respectively, such that A c 171, B C 172 and 71 cl Yl N72 cl V2 = ~ , there exists a partition T in X between A and B which satisfies the inclusion TNYCT'. If (Y, r~, 7~) is a p-closed BsS of a hereditarily p-normal BS (X, 71,72) and A C co 71, B C co 72, with A N B - ;g, then for every partition T' in (Y, 7~, 7~) between A N Y and B N Y, there exists a partition T in X between A and B such that T N Y C T'.
114
III. Dimension of Bitopological Spaces
Proof. By Corollaries 0.2.3 and 0.1.8, there exist V1 E ~-2, 1/2 r T1 such t h a t A c V1, B c V2 and 7"1 cl V1 n 7-2 cl V2 - ~ . O n the other hand since (Y, ~-~, ~-~) is p-normal, there are V1~ c T~, V~ c ~-~ such t h a t
Y N n cl Vl C V[, Y N~2cl V2 c V~ and V/ N VJ = ~. Let T' = Y \ (V{ U V~). It is obvious t h a t A N "r2 cl V~ = 2~ = B AT-1 cl VI'.
(1)
Let us consider the BsS (V1~ U V~, ~ ' , ~-~') where V{ E T~', V~ E r['. It clearly follows t h a t V1~ N Ts cl V~ - 2~ - V~ n rI' clVl t. (2) Therefore, by (1) and (2), we have Tl c l ( A U V;) n T2 c l ( B U VJ) = (A U T1 c l V ; ) n ( B U T2C1V~) =
and by T h e o r e m 0.2.2, there exist HI c T2, /-/2 c T1 such t h a t A U 17[ c HI, B U V~ c /-/2 and H i n / - / 2 z ~ . The set T = X \ (H1 U/-/2) is a partition in X between A and B such t h a t
T N Y = Y \ (H1 UH2) C Y \ (VI~U V~)= T' so t h a t we have proved the first part. N o w let (Y - T1 cl Y O ~-2 el Y, T~, ~-~) be a p-closed BsS of (X, T1, ~-2) and T' - Y \ (V[ U V~) be any partition between A N Y r co ~-~ and B N Y c co 7-~ so that A N Y C V; E T~, B N Y C V~ C T~ and V; N V~ = 2;. Let V1" e ~-~' in (~-2cl Y, ~-;', T~') and V~" r T~" in (T1 C1 Y, T;tt, 7"~tt)such t h a t A n 72 el Y c V~', B N 71 el Y c V~". If V1 E T2, V2 c ~-1 are sets for which V1 n 7-2 cl Y = V1" and 1/2 N f l c1Y = V~", then
A N (T2 c l Y \ V 1 ) = A N (T2 e l Y \ V~') = 2~ and B n (T1 c1Y \ V2) = B n (T1 c1Y \ V~tt) = ~ .
Thus
A c X \ (r2cl Y \ V1) = U c r2, B c X \ (rl cl Y \ V2) = V c rl. Since A N B = 2~, there exist G E 72, H E T1 such t h a t A c G, B c H and G N H = ~. Let M = U O G r 72, N = V O H c 71. T h e n A c M , B c N and by (4) of Proposition 0.1.7, there exist Q E T2, P c T1 such t h a t A c Q c T1 cl Q c U, B c P c T2 cl P c V and rl cl Q N 72 cl P = Z. It is evident t h a t TlclQ o Y C UNY
= Y\
(T2 c l Y \ V 1 ) = V;,
T2 c l P n Y C V n Y = Y \ (T1 c l Y \ V2) = V~ and so T ~ = Y \ (VII U V~) is the partition between T1 cl Q N Y and 7-2 cl P N Y. Therefore, there exists a partition T between A and B in X such t h a t T N Y C T ~. D
3.2. Pairwise Large Inductive Dimension
115
T h e o r e m 3.2.15. Let ( X , 7-1,T2) be a R-p-T1, d-second countable and p-normal BS, and Y C X , p - I n d Y - 0(<--> p - i n d Y - 0). Then for every pair of disjoint sets A c co T1 and B E co 7-2, there exists a partition T between A and B such that T N Y - 2~. Proof. Let us consider V1 c ~-2, I/2 c T1 such that A c V1, B c 1/2 and T1 cl V1 N ~-2 cl V1 - 2~. Then by Theorem 3.2.12 the empty set is a partition in Y between Y N 7-1 cl V1 and Y N T2 cl 1/72. Thus it remains to use Lemma 3.2.14 since by Corollary 0.1.13, (X, T1, T2) is p-perfectly normal and hence, be remark before Proposition 0.1.12, it is hereditarily p-normal. []
Of fundamental importance is
T h e o r e m 3.2.16 (A G e n e r a l i z e d Version of Dowker's A d d i t i o n Theorem). A s s u m e that for a hereditarily p-normal BS (X, T1,7"2) We have" X,~ c T1 N 7-2 f o r every m -
1,oc, X,~+I c
(i,j)-Inld(X,~ \ X,~+I) <_ n for each m -
X~,
X1 -
X
and
oo n x,~
m=l
1, oc, then ( i , j ) - I n d X
-
;g.
If
< n.
Proof. The theorem holds for n - - 1 . Let us assume that it holds for n - 1 and prove it for n. If A c coTj, B c coTi and A N B -- 2~, then by Corollary 0.1.8, there exist U0 c 7-~, V0 c Tj such that A c U0, B c V0 and Ty cl U0 N ~-i cl V0 - 2~. Let Dm - X,~ \ Xm+l for every m - 1, oc. We begin by constructing three mutually disjoint sets U1, C1, V1, where U1 is/-open in X1 - X, V1 is j-open in X1 - X,
C1 C D1 is p-closed in D1,
(i, j ) - I n d C1 < n -
1
and D1 C U 1 U C 1UV1,
Tj clU0 c U1, T~clV0 C 1/1, wjclU1 n ~-~clV1NX2 = ~.
Clearly, the sets rj cl U0 N D1 and r~ cl V0 5 D1 are disjoint in (D1,7-{, T~). Since (i, j ) - I n d D 1 < n, there exists a partition C1 in D1, corresponding to the pair (rj cl U0 N D1, ri cl V0 n D1) such that D1 \ C1 = (71 U H1, where (71 is/-open in D1, H1 is j-open in D1, 7-jclU0 ND1 C (71, TiclV0 N D 1 C H1 and G 1 N H 1 = 2~. Since D1 is simultaneously 1- and 2-closed in X1 = X and C1 is p-closed in D1, we conclude that C1 is also p-closed in X1 = X. On the other hand, D1 \ C1 = (X1 \ C 1 ) n D1 so that D1 \ C1 is simultaneously 1- and 2-closed in X 1 \ C1. But D1 \ C1 = G1 U H1, where G1 n H1 = 2~ and, hence, (71 is simultaneously/-open and j-closed in D1 \ C1, H1 is simultaneously j-open and /-closed in D1 \ C1. Therefore (71 is j-closed in X1 \ C1 and H1 is /-closed in X1 \ C1. Let A1 - (Tj el U0 U (71)n (X 1 \ C1) and B 1 - (T i C1V0 U H 1 ) n (X 1 \ C1).
It is obvious that (A1,B1) is a pair of disjoint respectively j-closed and/-closed subsets of X1 \ C1 which is p-normal because (X, T1, T2) is hereditarily p-normal. Therefore, there exists a n / - o p e n set U 1 and a j-open set V1 in (X 1 \ C1, T~t, T~r
116
III. Dimension of Bitopological Spaces
such t h a t A1 c U1, B1 C V1 and ~_~Icl U1 n 7-:I c1 V1 = ~ so t h a t 7-jclV 1 nT-~clV1 n (X 1 \ C 1) = ~.
Thus Tj cl U1 n ri clV1 n (Xl \ C1) :
and, as a result, Tj cl U1 N 7-~cl V1 C C1. Since C1 c D1 = X1 \ X2, we obtain rj cl U1 N ri cl V1 n X2 = ~. Furthermore, C1 -- D1 \ (G1 U H1) = (D1 \ G 1 ) n (D1 \ HI), where D 1 \ G 1 c co T~, D1 \ H1 C co Tj, and, hence, D1 \ G1 c co Ti, D1 \ H1 E co Tj since D1 E co T1 n co 7-2. Now, we have X1 \ C 1 - - X l
\ ((D1 \ G 1 ) n (D 1 \ H 1 ) ) -
- - ( X l \ (D1 \ G1)) U (X 1 \ (D 1 \ H1) ) - ( ( X 1 \ D1) U (~1) U ((X 1 \ D1) U H1). Since U1 C X1 \ C1 and U1 n H1 = ~, we obtain U1 C (X1 \ D1) U G1. But the fact that U1 is /-open in X1 \ C1 implies that U1 is /-open in X1 = X because (X1 \ D1) U G1 is/-open in X1 = X. In a similar manner it can be proved that V1 is j-open in X1 = X. Hence D1 = G1 U C1 U H1 and G1 C A1 c U1, H1 C B1 c V1
imply D1 c U1 U C1 U V1. Moreover, Tj cl U0 N D1 C G1 implies that Tj cI Uo N CI = K~ a n d s o
7j cl Uo = Tj cI Uo N (XI \ C1) c A1 c U1.
Similarly, ~-i cl 170 c V1. Thus for m = 1, we have constructed three mutually disjoint sets Urn, Cm, V.~ satisfying the following conditions: (l~ (2 ~ (3 ~ (4 ~
(5~ (6 ~
U.~ and V.~ are contained in X.~ and are respectively i- and j-open in X. C.~ is contained in Dm - X.~ \ Xm+l and is p-closed in D.~. c u u Tj cl Um n 7-~cl V,~ n X m + l - ~J.
clum_l a x m c
clvm_l
c
(i, j)- Ind Cm <_ n - 1.
Assume that we have constructed three mutually disjoint sets satisfying the above conditions for 1, 2 , . . . , m - 1 and let us construct them for m. Condition (4~ implies that Tj cl Urn-1 O ~-~cl V.~-I n Xm - ~. Recalling that Dm c X.~, that (Tj cl U.~_I O Din, "q C1V.~-I n D.~) is a pair of disjoint j- and/-closed sets in Din, and following the inequality ( i , j ) - i n d D ~ <_ n, there exists, in D.~, a partition C.~ corresponding to the pair above such that (i, j ) - I n d C.~ _< n - 1. To the partition C.~, there also correspond disjoint i- and j-open sets Gm and H.~ such that D.~ = G m U C.~ U H.~ and Tj cl U,~_ 1 n D.~ c G.~, ~-~cl V.~_ 1 n D.~ c H.~. Since C.~ is p-closed in D.~, and D.~ is simultaneously 1- and 2-closed in X.~, Cm is also p-closed in Xm. The sets G.~ and H.~ are j- and/-closed in D m \ C~, and since D.~ \ C.~ = (X.~ \ C.~) N D.~ and D.~ is simultaneously 1- and 2-closed in X.~, Dm \ C.~ is simultaneously 1-closed and 2-closed in Xm \ C.~. Hence G.~
3.2. Pairwise Large Inductive Dimension
117
is j-closed and Hrn is/-closed in Xrn \ Crn. Furthermore, it is easy to verify that the pair (Am, Brn), where
Am - (Tj cl Um-1U Grn) N (Xm \ C r n ) -
(7-j cl Um-1N (Xm \ Cm)) u am,
Bm - (Ti cl Vm_ 1 U H m ) N (Xrn \ Urn) - (7-/cl Vm_ 1 n ( a m \ Cm)) O H m is the pair of disjoint respectively j- and/-closed sets in X m \ C~. Since Xrn \ C~ is p-normal, there exist, in Xrn \ Cm, disjoint respectively i- and j-open sets Um and V,~ such that
Arn c U.~, B.~ C Vm and "rj cl Urn n "ri cl V.~ N ( X ~ \ Crn) -- ~ so that rj cl Urn N ~-i cl Vrn C Crn. Since C.~ c D.~ = Xm \ Xrn+l, we obtain ~j cl U,~ a r~ cl E~ a Xm+l = ~. The set Xrn \ C.~ is p-open in X.~, and since Xrn is simultaneously 1- and 2-open in X, Xrn \ C.~ is p-open in X so that X.~ \ C.~ = O1 U 02, where Oi c ri. Therefore, we obtain Urn c Oi, V.~ c Oj and thus Urn c ri, Vrn E rj. Furthermore, D.~ = G.~ U C.~ U H.~, and since Grn c Arn c U.~, H.~ c Brn c Vrn, the inclusion D.~ c U.~ U C.~ U Vm is correct. Moreover,
Tj C1 Urn-1 n Drn C Grn implies rj cl Urn_ 1 n Crn and, therefore,
Tj cl Urn-1 n Xrn - Tj cl Urn-1 n (Xrn \ Crn) C Arn c Urn. Similarly, ri cl Vrn-1NXrn C Vm. We have thus constructed three mutually disjoint sets Urn, Cm, Vm, satisfying conditions (l~176 oo
Let us now assume that C -
U Crn and prove that (i, j ) - I n d C < n - 1. Let r~--1
oo
Zrn -
U Ck for each m -
k=m
1, oc; in particular, Z1 - C .
Zrn-bl C Zm,
Also,
Z m \ Z m + l - Cm
so that
(i, j)- Ind(Zm \ Zrn+l ) ~ T t - 1. Since Xm N C - Zm, the set Zrn is simultaneously 1- and 2-open in C for each (2<) m
--
1, oc and n
Zm -- 2~. Thus by the inductive assumption (i, j)- Ind C <_ n - 1.
rn=l
Finally, we are to show that C contains a partition, corresponding to the pair
(A,B).
Let U -
c~
oG
U Urn, V -
U Vm. It is clear that U c ~-~ and V c rj.
m--1
m---1 oo
Condition (5 ~
implies that A c U0 c U, B c 170 c V. Also, X -
U Drn and, rn=l
u s i n g D m C Urn U Cm U Vm, we have X - U U C U V. Let us prove that U N V - 2~. Clearly, it suffices to show that Us N Vt - 2~ Ks N Ut. Indeed, s < t implies
Us n Vt c Us n X t C (Us n X s + l ) n X t
C gs+l n X t
and so Us N Vt C gs+l n l/t. Continuing this reasoning, we obtain Us N Vt C U t n V t - 2 ~ . By analogy, V s N U t - ~ s o t h a t UNV--2~. LetT-X\(UUV).
118
III. Dimension of Bitopological Spaces
It is obvious that T is the partition, corresponding to (A, B). Moreover, T C C is p-closed in C and, therefore, (i, j ) - I n d T < n - 1. [] C o r o l l a r y 3.2.17. Let there exist for a hereditarily p-normal BS (X, 7-1,7-2) an X,~ C 7-1 N ~-2 for every m -- 1, o0, such that X,~+I c X,~, X1 - X , and O0
n
xm - z. Ifp-Ind(Xm
\ Xm+t) <_ n for each m - 1, oo, then p - I n d X _< n.
rn= 1
C o r o l l a r y 3.2.18. Let (X, 7.1,7.2) be a hereditarily p-normal BS and A c co 7.1 N co 7.2. Then the following statements hold: (1) If ( i , j ) - I n d A < n and ( i , j ) - I n d ( X \ A) < n, then ( i , j ) - I n d X (2) /jr p - Ind A < n and p - I n d ( X \ A) < n, then p - Ind X < n. Proof. (1) Write X1 - X, X2 - X \ A , rem 3.2.16. Assertion (2) is obvious.
X3 - X4
....
< n.
0 and apply TheoD
The following fact is equivalent to the one proved above. T h e o r e m 3.2.19 9 Let (X, rl, r2) be a hereditarily p-normal BS and {Din} m~176 = l be a disjoint sequence of sets covering X such that Fs - O D,~ c co 7-1 n CO 7"2 m<s
for each s -
1, oo. Then the following conditions are satisfied:
(1) If ( i , j ) - I n d D , ~ <_ n for each m - 1, oc, then ( i , j ) - I n d X (2) If p - I n d D,,~ <_ n for each m - 1, oo, then p - I n d X < n.
<_ n.
Proof. If in Theorem 3.2.16 it is assumed that D,~ - Xm \ Xm+l, then we obtain Theorem 3.2.19. To prove the converse, it is enough to take X~ - X \ U Din. D m<8
T h e o r e m 3.2.20. If (X, 7-1,7-2) i8 a hereditarily p-normal BS and X - P u Q , where ( i , j ) - I n d P <_ n, ( i , j ) - I n d Q <_ O, then ( i , j ) - I n d X <_ n + 1. Proof. Let B be any j-closed subset of X and U ( B ) be its a n y / - o p e n neighborhood. Then there exist a j-open set V and a n / - o p e n set W such that A - X \ U ( B ) C V, B C W and 7.i cl V N Tj cl W - Z. It is obvious that Q \ (7.i cl V N Q) is a n / - o p e n neighborhood of the j-closed set rj cl W n Q in (Q, r~, r~). Since (i, j)- Ind Q _< 0, there exists a simultaneously/-open and j-closed set U in Q such that
- dwnQ
c u c Q \
d v n Q).
Let us consider the sets B U U and A U (Q \ U) and show that
Since 7j cl(B U U) - B U 7y cl U and 7~ cl (A U (Q \ U)) - A u 7.~cl(Q \ U), the proof of (,) reduces to the following facts"
d u n (Q \ U) (b) B N ( Q \ U ) - 2 ~ . (c) 7j cl U N A - ;g.
0 @
@ ;> q;
q; b.O r,,-i
d o4 /
II
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II
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II ~
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r
II ~
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u~ i =~c~ c
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II 1'~
9-
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120
III. Dimension of Bitopological Spaces
Proof. Let A c co Tj, B E co T~ and A A B - 2~. We shall show that to the pair (A, B) there corresponds an e m p t y partition. Since (i, j)- Ind X1 - 0, there exist a j-closed a n d / - o p e n set A1 and an/-closed and j-open set B1 in X1 such that
A n X~ c A1,
A1 U B1 - X1 and A1 G B1 - 2~.
B DI X 1 c B1,
Since X1 E co7-1ACO72, we have A1 E coTj and B1 E coT/. Hence (AUA1, B U B 1 ) is a pair of disjoint j-closed a n d / - c l o s e d sets in X. Since (X, 7"1,1-2) is p-normal, there exist a n / - o p e n set G1 and a j-open set H1 such that A U A1 C G1,
B LJ B1 C H1
and Tj cl G1 N Ti cl H1 -- ~ .
It is obvious t h a t X 1 C G1 U H1. If in our previous reasoning we replace the pair ( A , B ) by the pair (rj clG1, ri clH1), we can construct an /-open set G2 and a j-open set H2 such that X2 c G2 LJ H2, Tj cl G2 A Ti clH2 -- ~ and Tj cl G1 C G2, Ti clH1 C/-/2. Proceeding with our reasoning in a likewise manner, we obtain two increasing sequences of i- and j-open sets in X: G1 c G2 C " " C Gm C "" ,
Tj cl Grn C Grn+ 1,
H1cH2c...cH~
zi c l H m C Hm+l,
c...,
X 1 U X 2 U . . . U X m C Grn U H,~,
LetG-
oo
(2o
U G,~,H-
U Hm. S i n c e X -
m=l
7j cl Gm • 7-~cl Hm - ;g. (ND
UXm, wehaveX-GUH.
rn=l
On
m=l
the other hand, G,~H,~-2~,
G,~cGm+I
and H m c H , ~ + I
for each m - l ,
oo
imply G N H - o. It is obvious t h a t A C G and B C H. Thus (i, j)- Ind X - 0 .
[-]
C o r o l l a r y 3.2.24. Let (X, T1, w2) be a p-normal BS and {Xm} ~.~=1 be a seoo
quence of subsets of X such that X -
U x m , x m cCOWlnCOT2, a n d p - I n d X m - O m=l
for each m -
1, oc. Then p - Ind X -
0. oo
Note t h a t Theorem 3.2.23 also holds if X,~ -
U Fm where F ~ E co 7-1 ACO n ,
7-2
n=l
and (i, j ) - I n d F ~ - 0
for each m -
1, ec, n -
1, oc. oo
C o r o l l a r y 3.2.25. Let (X, ~-1, ~-2) be a p-normal BS and Y -
U F,~, where rn=l
frn C c o
7-1 N c o 7-2
for each m -
1, oc. Then the following implications hold:
(1) ( i , j ) - I n d X - 0 ~ (i,j)-IndY (2) p - Ind X - 0 ~ p - Ind Y _< 0.
<_ O.
oo
Proof. (1) By assumption Y -
U Fro, where F,~ c co T1Aco 72 for each m - 1, oc. rn=l
Hence every F,~ is p-closed in X and by (2) of Proposition 3.2.7, (i, j)- Ind Fm _< 0 for each m - 1, oc. Therefore, by Theorem 3.2.23, (i, j ) - I n d Y _< 0. D Moreover, we obtain
3.2. Pairwise Large Inductive Dimension
Theorem
121
If a d-second countable and hereditarily p-normal BS of p-closed sets,
3.2.26.
( X , 7-1,72) can be represented as a union of a sequence F1, F 2 , . . .
where (1, 2 ) - I n d F ~ - 0 ( ~
for each n -
(2, 1 ) - I n d F ~ - 0 ~
p-IndF~
- 0)
1, oc, then (1, 2 ) - I n d X - 0 ( ~
(2, 1 ) - I n d X - 0 ~
p-IndX
- 0).
Proof. Let A c co T1, B C co T2 and A N B - ~. We shM1 prove that, there exist G c T2, H C T1 such t h a t AcG,
BcH,
GNH-;g
and G U H - X
(1)
and so the e m p t y set is a partition between A and B. By Corollary 0.1.8, there exist U0 c 72, V0 c 7-1 s u c h t h a t A c U0,
B c V0 and 7-1 cIU0 NT-2clV0 - Z .
(2)
We shall define inductively two sequences of 2-open and 1-open sets U0, U1,. 9 9 and V0, V1,..., respectively, satisfying for each k - 0, oc the following conditions" U k -1 C U k ,
Vk _ l C Vk if k _> 1, and 7-1clUk
N T-2 c l V k
--
;g ,
Fk C Uk U Vk, where F0 - 2~.
(3) (4)
Clearly, the sets U0 and V0, defined above, satisfy both conditions for k - 0. Assume t h a t the sets Uk and Vk, satisfying (3) and (4), are defined for k < p. If Fp -- T 1 c1Fp N 7-2 cl Fp, then the sets
7-2ClVp_1 n F p C c o
7-1 cl Up_ 1 n f p C co 7-1' a n d
7"2'
in (Fp,7-~,7-~) are disjoint. Since ( 1 , 2 ) - I n d F p - 0 by virtue of T h e o r e m 3.2.12, there exists a subset V c 7-~ n co 7-~ such t h a t 7-1cI Up_I N Fp C V
and
7-2 cl Vp_l N rp C Fp \ V.
Since 7-1 cl V c 7-1 CI ~ ,
7-2 cl(Fp \ V) c 7-2 cl Yp and Fp - 7-1 c 1 F p n 7-2 cl Fp,
w e have 7-1 C1V n T2 c l ( F p \ V ) - ~ .
Let C -- 7-1 cl V \ 7-2 cl Vp_l and D - 7-2 cl(Fp \ V) \ 7"1 cl Up_ 1. Then c v D) u (C n
cl D) -
Therefore, by T h e o r e m 0.2.2, there exist U c 7-2, W c 7-1 s u c h t h a t C c U, D c W and U N W - 2 ~ . It is evident t h a t V N W - - 2 J , ( F p \ V ) N U - ~ and, consequently, 71CI V N W - ~ , 7-2 cl(Fp \ V ) N U - ;~. Let (I)l -- 7-1c1Up-1 U ( 7 - 1 c l V \ H/r),
(I) 2 - 7-2c1Up- 1 U (7-2cl(Fp \ V ) \ U ) .
Then ~i c coT-i and ~ 1 N ~ 2 -- 2~. Since (X, 7-1,7-2) is p-normal, there exist Up c 7-2, Up C 7-1 such t h a t ( ~ 1 C Up~ ~2 C Up and 7-1c1Up n T-2 cl Vp - (2~.
122
III. Dimension of Bitopological Spaces
T h e sets {Up}~_ 1 and {Vp}p~__l satisfy (3) and (4) for k - p . T h u s the construction of the sequences U0, U 1 , . . . and V0, V1,... is completed. It follows from (2), (3) and (4) t h a t the unions G -
oo
o(3
U Up and H -
U Vp satisfy (1).
p=l
D
p=l
Corollary 3 . 2 . 2 7 . I f a d-second countable and hereditarily p-normal BS (X, T1 < ~-2) can be represented as a union of a sequence F 1 , F 2 , . . . of i-closed sets, where p - Ind F~ - 0 for every n - 1, oc, then p - Ind X - 0. C]
Proof. T h e proof is obvious because co rl c co 72 c p - d l ( X ) .
Corollary 3.2.28. If a R-p -T1, d-second countable, and p-normal BS ( X , T 1 , T 2 ) can be represented as a union of a sequence F 1 , F 2 , . . . of p-closed sets where p - ind F~ - 0 for every n - 1, oc, then p - ind X - 0. Proof. By Corollary 3.2.13, p - I n d F ~ = 0 for each n = 1, oc. On the other hand, by Corollary 0.1.13, (X, 7"1,7-2) is p-perfectly normal and, hence, it is hereditarily p-normal. Therefore, by T h e o r e m 3.2.26, p - I n d X = 0. It remains to use once more Corollary 3.2.13. [3
Corollary 3.2.29. If a (R-p-T1) d-second countable, and hereditarily p-normal (p-normal) BS (X, T1, T2) can be represented as a union of a sequence F 1 , F 2 , . . . , where every Fk is a countable union of p-closed sets, that is, Fk = O(3
U F~, and p - I n d F k = 0 ( p - i n d F k = 0) for each k = 1, oc, then p - I n d X
= 0
n--1
( p - i n d X = 0).
Pro@
x-u
By Corollary 3.2.8, p - I n d F ) O(3
= 0 for every k = 1, co, n = 1, ec and
OO
ur .
k=l n=l
For p - i n d X it remains to use (2) of Proposition 3.1.4 in conjunction with Corollary 3.2.13. D
Corollary 3.2.30.
If a R - p - T 1 ,
d-second countable,
and p-normal BS
( X , T1 < 7-2) can be represented as a union of two BsS's Y and Z, where p - I n d Y -
p - Ind Z - 0 and one of them is l-open, then p - Ind X - 0.
Proof. Let, for example, Y E rl. T h e n X \ Y r co 7"1 C CO T2 C
p-Cl(X), X \
Y c Z
and, by Corollary 3.2.8, p - I n d ( X \ Y) - 0. Moreover, (X, T1 < 72) is p-perfectly oo
U Fk, where Fk E cot2 C p - e l ( X ) .
n o r m a l and so Y E 2 - 5 ~ ( X ) , t h a t is, Y -
k=l
By (2) of Proposition 3.2.7, p - I n d Fk - - 0 for every k -
1, oc. Therefore
O(3
X-YU(X\Y)-
UFkU(X\Y)' k=l
and it remains to use Corollary 3.2.27.
[3
T h e o r e m 3 . 2 . 3 1 . For every R-p-T1, d-second countable, and p-normal BS (X, T1,7-2) we have p - ind X - p - Ind X .
3.2. Pairwise Large Inductive Dimension
123
Proof. By (3) of Proposition 3.2.7, it suffices to prove only t h a t p - I n d X _< p - i n d X . It is evident t h a t one can assume t h a t p - i n d X < oc. We shall apply induction with respect to p - i n d X . Let p - i n d X - 1. Then, by T h e o r e m 3.1.29, fox" every disjoint pair of sets A c co 7-1 and B c co 72, there exists a partition T between A and B such t h a t p - i n d T _< 0. Hence, by Corollary 3.2.13, p - I n d T _< 0 and so p - I n d X < p - i n d X. Let us assume t h a t this inequality is also correct for k <_ n 1 and prove it for k - n. Let A c cot1, B c cot2 and A N B - 2~. Then by T h e o r e m 3.1.29, there exist a partition T between A and B such t h a t p - i n d T <_ n - 1. It follows from the inductive assumption t h a t p - I n d T < n - 1. Therefore p - I n d X < n and so p - I n d X <_ p - i n d X. [] C o r o l l a r y 3.2.32. If a R-p-T1, d-second countable, and p-normal BS (X, ~-1,T2) can be represented as a union of a sequence F1, F2,... of p-closed BsS's such that p - Ind Fk < n for each k - 1, ~ , then p- Ind X < n.
Proof. It is evident t h a t p - i n d Fk < n for each k - 1, cx~ and by Corollary 3.1.23, p- ind X <_ n. Thus p- Ind X < n. D C o r o l l a r y 3.2.33. If a 1-T1, d-second countable, and p-normal BS (X, T 1 < 7-2) can be represented as a union of a sequence F1,F2,... of i-closed sets, where p - I n d Fk < n for each k - 1, cx~, then p - Ind X < n. C o r o l l a r y 3.2.34. If a R-p-T1, d-second countable, and p-normal BS can be represented as a union of a sequence F1, F2,..., where every
( X , T1,7-2)
(N)
Fk is a countable union of p-closed sets, that is, Fk -- U F~ and p - Ind Fk < n m=l
for each k - 1, oc, then p - Ind X < n. Pro@ Since p - I n d F k < n and F~ c p-Cl(X) for each m - 1, oc, by (2) of Proposition 3.2.7, p - I n d F ~ <_ n for each k - 1, oc and m - 1, oc. Thus it remains OO
to use Corollary 3.2.32 as X -- U
(X)
U Fmk"
[-1
k=lrn=l
C o r o l l a r y 3.2.35. If a 1-2/71, d-second countable, and p-normal BS (X, rl < 72) can be represented as a union of two BsS's Y and Z, where p - I n d Y < n, p - I n d Z < n, and one of them is 1-open, then p - I n d X < n.
Proof. Let Y c
T1.
Then T1 C 2-2~'cr(X) since (X, T1,T2)is p-perfectly normal.
OO
Now Y -
U Fk, where Fk C coT2 C p-Cl(X) for each k -
1, oc and by (2) of
k=l
Proposition 3.2.7, p - I n d Fk _< n for each k - 1, oc. On the other hand, X\YcZ,
X\YEcoT-1cp-Cl(X)
and by the same reasoning, p - I n d ( X \ Y) _< n. But (X)
X - Y U ( X \ Y) - U Fk U ( X \ Y) k=l
and it remains to use Corollary 3.2.32.
D
124
III. Dimension of Bitopological Spaces
T h e o r e m 3.2.36. Let (X, T1,T2) be a p-normal BS, let n denote a nonnegative integer and let (71,72) has on X the (i,j)-(coTj,A)-insertion properties (,~->, (7-1,72) has on X the ( i , j ) - ( 2 x , A ) - i n s e r t i o n properties), where A {V c Ti" (i, j ) - Ind(j, i)- Fr V _< n - 1}. Then (i, j)- Ind X _< n. This theorem can be proved by repeating the scheme of the proof of Theorem 3.1.34, taking into account (4) of Proposition 0.1.7 and Proposition 2.4.10. C o r o l l a r y 3.2.37. Let (X, T1, 7-2) be a p-normal BS and (71,7-2) has on X the (i,j)-(covj,T~ C~ covj)-insertion properties (z----> (T1,T2) has on X the (i, j)-(2 x , Ti N COTj)-insertion properties). Then (i, j)- Ind X - 0. 3.2.38. The following conditions are satisfied for a BS (X,7-1,7-2)" (1) If 7-1
Theorem
Proof. (1) It is clear that the inequality 1- Ind X <_ (1, 2)- Ind X holds for ( 1 , 2 ) - I n d X - oc. Let us assume that ( 1 , 2 ) - I n d X - k < oc and show t h a t 1-Ind X _< k. For k - - 1 the required inequality is obvious. Now let us assume that the inequality is correct for k < n - 1 and prove it for k - n. Since (1, 2 ) - I n d X - n, for every 2-closed set, in particular, for every l-closed set F and its any 1-open neighborhood U ( F ) , there exists a 1-open neighborhood V ( F ) such that 7-2 cl V ( F ) C U(F) and (1, 2)-Ind(2, 1)- Fr V ( F ) < n - 1. Since T1 < C 7"2 and V ( F ) 9 T1, by (3) of Corollary 2.2.7, wi cl V ( F ) - - T1 cl V ( F ) and thus (2, 1)- Fr V ( F ) - 1- Fr V ( F ) . Hence (1, 2)-Ind(1- Fr V ( F ) ) <_ n - 1 and by the inductive assumption 1- Ind(1- Fr V ( F ) ) _< n - 1, t h a t is, for every 1-closed set F and its any 1-open neighborhood U(F) there exists a 1-open neighborhood V ( F ) such that T1 cl V ( F ) C U(F) and 1-Ind (1- Fr V ( F ) ) <_ n - 1. Therefore 1-Ind X < n. (2) The first inequality is obvious by (1) and Corollary 2.3.10. Furthermore, by analogy with the above reasoning, we can prove that (2, 1 ) - I n d X _< 2 - I n d X . Let us assume that the inequality holds for 2 - I n d X - k _< n - 1 and prove it for k - n. Since 2 - I n d X - n, for every 2-closed set, and in particular, for every 1-closed set F and its any 2-open neighborhood U(F) there exists a 2-open neighborhood V ( F ) such t h a t ~-2 el V ( F ) C U(F) and 2-Ind(2- Fr V ( F ) ) <_ n - 1. Since ~-1
1. 89
3.3. Pairwise Covering Dimension
125
3.3. Pairwise Covering D i m e n s i o n As in [106, p. 472], by the order of a family A of subsets of a set X we mean the largest integer n such t h a t the family A contains n + 1 sets with the nonempty intersection or oc if no such integer exists. Thus if the order of the family A - {As}s~s is equal to n, then we have Asl n As~ n . . . N As,~+~ - ;g for any n + 2 members As~, A ~ 2 , . . . , As,~+2 of A. In particular, a family of order - 1 consists only of an e m p t y set, while the family of order 0 consists of pairwise disjoint sets not all of which are empty. The order of a family A is denoted by ordA. D e f i n i t i o n 3.3.1. Let (X, T1,7"2) be a BS and n denote a nonnegative integer. We say t h a t (1) ( i , j ) - d i m X - - 1 ,z-----5, X - ;g. (2) ( i , j ) - d i m X <_ n if for every family of /-open sets {Us}sm=l aIld every family of j-closed sets {Fs } ~ 1 together with Fs C Us for each s - 1, m, there exists a family o f / - o p e n sets {Vs}~=l such t h a t Fs C Vs c Us for each s - 1, m, and ord{ (j, i)-Fr Vs } L 1 -< n. (3) ( i , j ) - d i m X - n if ( i , j ) - d i m X <_ n and the inequality ( i , j ) - d i m X <_ n - 1 does not hold. (4) (i, j ) - d i m X - o c if the inequality (i, j ) - d i m X _, ((1, 2 ) - d i m X < n A (2, 1 ) - d i m X < n).
P r o p o s i t i o n 3.3.2. BsS ( Y , T ~ , 4 )
W e have ( i , j ) - d i m Y <_ ( i , j ) - d i m X of a US ( X , T1,7-2).
f o r every p - c l o s e d
P r o @ It ~umces to show t h a t (i, j ) - d i m X - k implies (i, j ) - d i m Y < k. The inequality is correct for k - - 1 , k - oc. Let us assume that it holds for k _< n - 1 m and prove it for k - n. We assume that {U~ } L 1 and {F~! }s=l are respectively families o f / - o p e n and j-closed subsets of (Y, T~, T~) such that F~ C U~ for each s - 1,m. Consider the families {Us}~__l and {Fs}~=l o f / - o p e n and j-closed sets, respectively, in (X, T1,7-2) such that U~ - U s N Y and F~ - F s N Y for each s - 1, m. Since Y - Y1 n Y2, where Yi c co Ti and U - X \ Y~ C Ti, the families {E s 9 m s --
U s
U U}sm=l and {(I)s 9 q)s - Fs N Yj }srn=l
consist respectively of/-open and j-closed subsets of X such t h a t (I)s C Es for each s-l,m.
By virtue of the equality ( i , j ) - d i m X /-open sets in X such t h a t (I)s c V~ c Es for each s -
-
n, there exists a family {Vs}L1 of
1 , m and o r d { ( j , i ) - F r V s }
Let us consider the family o f / - o p e n sets {V~" V / -
"~ s=l -- n.
Vs N Y}~--1 in Y. Then
E~ n Y - (Us U U) N Y - (Us N Y) U (U N Y) - Us n Y, (I)s N Y - (Fs N Yj) N Y -- (Fs n Yj)N (Y1 N Y2) -Fs N (Yl N Y 2 ) - F s N Y - F '
s
126
III. Dimension of Bitopological Spaces
for each s - 1, m. Therefore F~ C V~ c Us' for each s - 1,m. It remains for us to prove t h a t ord{(j, i)- Fry V~}~__1 _< n so that, it suffices to show t h a t (j, i)- Fry V~C(j, i)- Fr Vs for each s - 1, m. But the latter inclusion is obvious and, therefore, (i, j ) - d i m Y<_ n. [-I C o r o l l a r y 3.3.3. We have p - d i m Y (Y, ~-~,~-~) of a BS (X, T1,T2).
_< p - d i m X
for every p-closed BsS
T h e o r e m 3.3.4. The equalities (i, j ) - d i m X - 0 and (i, j)- Ind X - 0 are equivalent for every BS (X, T1, ~-2) and either of them yields the p-normality of
(X, T1, T2). Pro@ We begin by assuming t h a t ( i , j ) - d i m X - O , A c coT j, B c coT i and A A B - ~. Then U - X \ B is t h e / - o p e n neighborhood of A. Since (i, j ) - d i m X - 0, there exists a n / - o p e n set V such t h a t A c V c U and ord(j, i ) - F r V _< 0 so t h a t (j, i)- Fr V - ~ and, therefore, V E Ti C~co Tj. Hence the set
W--X\VcTjAcoTi,
AcV,
BcW
and V N W - - ; g
so t h a t (X, T1, ~-2) is p-normal. Moreover, V O W - X implies t h a t the e m p t y set is the partition in X, corresponding to the pair (A, B) and thus (i, j ) - I n d X - 0. Conversely, assume t h a t ( i , j ) - I n d X - O. Then by (1) of Proposition a.2.7, (X,~-1,72) is p-normal. Let {Us}s-1 and {Fs}srn=l be respectively families of /-open and j-closed subsets of X such t h a t Fs c Us for each s - 1, m. Since ( i , j ) - I n d X - O, we have a family of simultaneously /-open and j-closed sets { V s } ~ l together with F~ c Vs c Us for each s - 1,m. Thus
(j,i)-FrV~ - ~ g for each s -
1, m a n d s o
ord{(j,i)-FrVs}s~__l- -1.
Hence (i, j ) - d i m X <_ 0, and X # 2~ implies t h a t (i, j ) - d i m X - 0. D Note t h a t due to Proposition 3.2.11 the following equivalences hold for a BS
(X, T1, T2)" (1,2)-dim X - 0 ,z---> (2, 1)-dim X - 0 ,z---->,p_dim X - 0. C o r o l l a r y 3.3.5.
The following conditions are satisfied for a BS (X, ~-1, ~-2):
(1) p -dim X = 0 ~
p - Ind X = 0 and either of them yields the p-normality
of (X, Tl, 72). (2) If ( X, T1, ~-2) is j-T1, then
( i , j ) - d i m X - 0 ( ,z---->, ( i , j ) - I n d X
- O) ---5, ( i , j ) - i n d X - 0
so that, if (X, 7-1, ~-2) is R-p-T1, then p - d i m X - 0 ( ,z---->,p - I n d X
- 0) ----5, p - i n d X
- 0.
Proof. The condition (1) is obvious. Assertions of (2) are immediate consequences of (3) of Proposition 3.2.7. C o r o l l a r y 3.3.6.
The following statements hold for a BS (X, T1, T2)"
D
3.3. Pairwise Covering Dimension
127
(1) I f (X, T1,T2) i8 p-normal and {Xm}m~=l is a sequcTtec of subsets of X such that CO
X -- U Xrn~ Xm C COT1 rlCO7-2 and ( i , j ) - d i m X m - 0 m=l - 0 so that if p - d i m X , ~ - 0 f o r
f o r each m - 1, oc, then ( i , j ) - d i m X each m - 1, oc , then p -dim X - 0. (N:)
(2) If (X~ T1, 7"2) i8 p-nof~Ttal and Y -
U F,~, where Fm C co~-i ~ coT2 f o r m=l
each m - 1, oc , then (i,j)-dimX
- 0 ~
p-dimX
- 0 ~
(i,j)-dimY
<_ 0
and so
p - d i m Y < O. n
(3) I f ( X , 7-1, T2) is hereditarily p - n o r m a l and X -
[_J X . ~ , then rrt=O
((i, j ) - dim X.~ <_ 0 f o r each m - O, n ) ---->. (i, j ) - Ind X < n and so
(p- dim X,~ < 0 f o r each m - O, n) -->, p - Ind X <_ n. Proof. (1) and (2) are immediate consequences of Theorem 3.2.23 and Corollaries 3.2.24 and 3.2.25 in conjunction with Theorem 3.3.4. Assertions of (3) follow directly from Corollary 3.2.22 and Theorem 3.3.4. D
R e m a r k 3.3.7. It is clear t h a t each of the nine dimension functions introduced here is an invariant of a d-homeomorphism (i.e., is a bitopological property). Moreover, their values coincide with integer 1 for the natural BS (R, c~1, c~2).
CHAPTER
IV
Baire-Like Properties of Bitopological Spaces A TS is said to have the Baire property if any first category subset has an empty interior in the space. This property was proved for the first time and independently by W. Osgood for the real line [197] and by R. Baire for spaces R ~ [20]. Today it plays an important role in analysis, topology and mathematical logic (see, for example, [63], [227], [199], [211]). Following N. Bourbaki, a Baire space is a TS such that every nonempty open subset is of second category [38]. Thus a Baire space is a space for which the Baire category theorem is true. A brief but cognitive story of the origin and development of the theory of Baire spaces as well as of its applications can be found in [173]. Most of the results on Baire spaces are stated in [133]. It is well-known that a subset A of a TS (X, w) can be of one category in (X, T) and of another category in itself as a subspace of (X, T), while for open subsets of (X, T) these categories coincide. This observation is the principal factor in defining Baire spaces in different equivalent ways. However, as illustrated by Example 1.5.8 and unlike the topological case, a n o n e m p t y / - o p e n subset of a BS (X, T1, ~-2) can be of one (i, j)-category in (X, Wl, 72) and of another category in itself as a BsS of (X, T1,T2). This argument is closely connected with the definition of (i,j)-Baire spaces [92] and serves as a good introduction to our further discussion. We give a general definition of (i,j)-Baire spaces which is based on the (i,j)-category requirement on nonempty /-open sets, though (1,2)-Baire spaces ( X , T 1 < 7-2) also have other descriptions not based on the (1, 2)-category requirement on nonempty 1-open sets. To illustrate this situation it is sufficient to recall the Slobodnik property [241], which is one of the Baire-like properties from [173] (the Baire-type properties in terms of [100]). However, as compared with the topological version of Bourbaki or our method, the description of (i, j)-Baire spaces in terms of the (i, j)-categories has greater subtlety. In order to investigate all the introduced bitopological modifications of Baire spaces, along with (i,j)-Baire spaces, we define the so-called almost-(i,j)-Baire spaces for which n o n e m p t y / - o p e n sets must be of (i,j)-second category with respect to the whole space. The merit of our definition lies in its versatility, which is clearly seen for a BS (X, T1 < w2) because in this case, as is natural of topology, the notions of a (1, 2)-Baire space and of an almost-(1, 2)-Baire space turn out to be equivalent. Virtually in every case, we consider T1 C 7"2. Sections 4.1 and 4.2 contain a thorough investigation of (1, 2)-Baire spaces and almost-(2, 1)-Baire spaces, respectively. In [61] G. Choquet gives an interesting description of Baire spaces by 128
4.1. (1, 2)-Baire Spaces
129
means of the so-called sifter on a TS. Generalizing the latter notion, we obtain a new characterization of almost (i, j)-Baire spaces. Section 4.3 presents the results on (i,j)-Baire spaces in a strong sense, that is, on BS's whose every nonempty /-closed BsS is of (i, j)-second category. The chapter ends with the definitions and study of other possible modifications of Baire spaces. Namely, the (1, 2)-strict, (2, 1)-weak, 2-weak, and 1-strict Baire spaces are introduced and investigated together with their interrelations and relations to the notions from the preceding sections. We would like to remark that the p-Baire BS's introduced by C. Alegre, J. Ferrer, and V. Gregori in [9] are just the p-Baire BS's from [92], that is, are the almost p-Baire spaces from [95], [100]. An overwhelming majority of the results of [9] were obtained in [92], [95]. Moreover, the notion of a pairwise fine BS in [9] is none other than a BS with S-related topologies studied and used for different purposes in [252] and [95] as well. Among the other results from [9] the interesting characterization of a special class of real normed lattices by means of p-Baire (in our terms, almost p-Baire) spaces should be noted.
4.1. (1, 2)-Baire Spaces Definition 4.1.1. An (i,j)-Baire space (briefly, (i, j)-BrS) is a BS (X, T1,T2) such that every nonempty/-open subset U of X is of (i, j)-second category. This definition immediately implies that if (X, T1, T2) is an (i, j)-BrS, then X is of (i, j)-Catg II. E x a m p l e 4.1.2. The natural BS (R, aJl,CU2) is an (i, j ) - B r S since for every set U c czi \ {2~}, the BsS (U, cz~,cz~) contains no nonempty (i,j)-nowhere dense sets. By (ii) of Theorem 1.1.3 in [laa], it is also clear that (R, CJl,W2) is an i-BrS. Therefore, to counterbalance Remark 2.1.3, the BS's (R,C~l) and (R,a~2) are both BrS's, but c~lScZ2 is not correct.
Theorem 4.1.3. The following conditions are satisfied for a BS (X, T1 < 7"2): (1) (X, T1,T2) is a (1,2)-BrS ~ (X, T1,T2) is a 1-BrS. (2) (X, T1,T2) i8 a 2-BrS > (X, T1,T2) is a ( 2 , 1 ) - B r S .
Moreover, we have for a BS (X, T1 <S 7-2)" (3) (X, T1,T2) is a (1,2)-BrS ~ (X, T1,T2) is a 1-BrS ~ 2-BrS ~ (X, 7-1,T2) is a (2, 1)-BrS.
(X, T1,T2) is a
Pro@ (1) By Corollary 1.5.14, if U c T1 \ {~}, then U is of (1, 2)-Catg II ~
U E (1, 2)-Catgli(X)
and, hence, by (7) of Theorem 1.1.24,
U C 1-Catgli (X) ~
U is of 1-Catg II.
(2) If U c T2 \ {2~}, then by (7) of Theorem 1.1.24, U is of 2-Cat9 II implies that U is of (2, 1)-CatgII since T~ C T~ in (U, T~, T~).
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134
IV. Baire-Like P r o p e r t i e s of Bitopological Spaces c~
1, oc. Hence, by (1)ofTheorem 1.1.24, U ( X \ A n ) 9 (1, 2)-Catgi (X).
for each n -
n=l
Following (4) of Theorem 4.1.6, (X, T1,T2)is also an A-(1,2)-BrS and by (3) of Theorem 4.1.4 in conjunction with Definition 4.1.5, oo
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oo
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and
N Ao - x n=l
so oo
O A~ e 2 - ~ ( x ) n 1-D(x). n--i
(3) The equalities are given by (3) of Theorem 2.2.20. For the inclusion assume that there exists a set A 9 2-Ga(X) N (2, 1)-Catgx(X ) such that T2 int T1 clA ~: ~. Hence, by (3) of Corollary 2.2.7, T1 int T1 cl A ~ ~ so that there exists a set V 9 T1 \ {~} such that V C T1 el A. Furthermore, V = V O T1 cl A c T1 cl(V N A) and thus V N A 9 1-:D(V). Clearly, A 9 2-Ga(X) implies that O(3
V\A-
OO
V \ n An - U ( V \ A n ) n=l
n--1
OO
U (v\ (VNAn)) 9 2-~(V) n--1
since every set V \ (V Cl An) is 2-closed in (17, 7-~,~-~), and ~-; int(V \ A) - r; int (V \ (V N d)) - V \ r; cl(V N A) as V Cl A 9 1-2)(V). Therefore, by (2)of Theorem 1.1.24,
v \ A e 2-7~(v) n 1-~d(V) c (1, 2)-Catg, (V). Following the condition, A 9 (2, 1)-Catgi(X ) and by (7) of Theorem 1.1.24, A 9 (1, 2)- datgx (X). Since V 9 71, by Corollary 1.5.9, A N V 9 (1, 2)- CatgI (V) and thus by (1) of Theorem 1.1.24, V = (V \ A) U (V N A) 9 (1, 2)-Catg x(V) so that V is of (1, 2)- Catg I, which contradicts V 9 T1 \ {;~} and (X, T1,T2) is a (1, 2)- BrS. Kl T h e o r e m 4.1.10. Let (X, rl < r2) be a (1,2)-BrS. If A c X is any subset, then A 9 (1,2)-Catgx(X) if and only if X \ A contains a 1-dense 2-G6-subset of X.
Proof. First, let (x)
A 9 (1, 2)-Catgx (X), that is A -
U An, AN 9 (1,2)-A/'2)(X) n=l
for each n = 1, oo. Hence {X \ 7-2cl An}n~1761 is a countable family of subsets of X such that X \ ~-2clAn 9 72A 1-2)(X) for each n - 1, o0. Using (2) of Theorem 4.1.4 and the first equivalence in (4) of Theorem 4.1.6, we obtain cx)
O (x \ ~ clA~) c 2-~6(x) n 1-D(X). n=l
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136
IV. Baire-Like Properties of Bitopological Spaces
C o r o l l a r y 4.1.12.
The following conditions are satisfied for a (1,2)-BrS
( X , TI
(1) I I A c 2-(~5(X)O 1-/)(X), then (A,~-~,T~) is a (1,2)-BrS. (2) If A c (1-Gs(X)N2-~9(X))U(2-Gs(X)N2-1)(X))O(1-Gs(X)N1-7)(X)), then (A,w{,T~) is a (1,2)-BrS.
Proof. (1) Suppose that A c 2-Ga(X)N1-D(X). Then X \ A c 2 - ~ ( X ) O 1 - B d ( X ) and by (2) of Theorem 1.1.24, X \ A c (1,2)-Catgi(X). Hence it remains to use Theorem 4.1.11. Assertions of (2) follow directly from (1), taking into account the inclusion TI C T2.
D e f i n i t i o n 4.1.13. A family of sets II = {Us }scs c T~ is an (i, j)-pseudo-open covering of a BS (X, T1,7"2) if Tj cl U Us = X. sCS
Note that the disjoint bitopological sum of a family of BS's {(Xs, T~, T~)}sCS is defined as a BS ( E Xs, T1, T2) where ( E Xs, Ti) is a disjoint topological sum sCS
sES
of the family of TS's {(Xs, 7-~)}scs. Later we use the abbreviation (i,j)-BrsS for an (i,j)-Baire subspace. T h e o r e m 4.1.14. The union of any family of l-open (1, 2)-BrsS's of a BS (X, T1 < T2) i8 a (1,2)-BrS and the union of any family of 2-open (1, 2)-BrsS's of a BS ( X , T1 < N 7"2) i8 a (1,2)-BrS.
Proof. First, let t ] = {Us}scS be any family of 1-open (1, 2)-BrsS's of (X, T1 < 72) and let U - U us be no (1,2)-BrS so that there exists a set V c T~ \ { ~ } in sCS
(U,T~,T~) such that V c (1, 2)- Catgi (U). Since V # ~, there is a set Us c i l together with V N Us # ~ so that V N Us c T~ \ {~} in (Us,~-~,T~). Since V C (1, 2)-Catgi (U) and Us c T~ \ { ~ } , by Corollary 1.5.9, we obtain that VNUs C (1, 2)-Catgi (Us) which is impossible because (Us, 7-~,T~)is a (1, 2)-BrS. Secondly, assume that il = {Us}soS is a family of 2-open (1, 2)-BrsS's of (X, T1 < N 7"2) and U = U us is not a (1,2)-BrS so that there exists a set sES
V c ~-~\{2~} in (U, T{, T~)such that V c (1, 2)-Catgi (U). Since V -r 2~, there exists a set Us c tl together with V a Us -r ~, and so V N Us c ~-{ \ {~} in (Us, ~-~,T~). By (2) of Corollary 2.3.13, ~-~
V n Us c (1, 2)-Catg I (U) = 2-Catg I (U). Following the well-known topological fact,
V
N
Us C 2-Catg I(U) ~
V
N
Us ff 2-Catg I(Us)
so that V N U s c 7-~ \ {~} and VNUs E (1,2)-Catgi(Us) , which is impossible since (Us, Tf, r~) is a (1, 2)- BrS. [-] C o r o l l a r y 4.1.15. The following conditions are satisfied for a BS (X, 7-1 < T2)"
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138
IV. B a i r e - L i k e P r o p e r t i e s of B i t o p o l o g i c a l S p a c e s
(3) follows via (1) of Theorem 4.1.3 and Proposition 1.30 in [133]. Assertion (4) is an immediate consequence of (1) of Theorem 4.1.3, Proposition 1.31 in [133] in conjunction with inclusion after Definition 1.1.21, and the inclusion T1 c 7-2. The proof of (5) follows directly from the necessary part of (i) in Theorem 1.24 in [133], taking into account (1) of Theorem 4.1.3 and the inclusion (1, 2 ) - S / ) ( X ) c Assertion (6) is an immediate consequence of (5). (7) Indeed, let A c X \ Y and A E 1-~5(X). Then, by (6), A E 1-H/9(X) and since X \ Y c 1-/)(X), by the well-known topological fact and Corollary 1.1.5, A r 1-N'D(X \ Y) c (1, 2)-N'D(X \ Y). [3 T h e o r e m 4.1.18. Let (X, T 1 < 7"2) be a (1,2)-BrS and (Y, ~-~,~-~) be a BsS of X . If Y c 1-Z)(X) and every set A c 1-81P(X)c~ 2 - G h ( X ) i n t e r s e c t s Y, then (Y, 7-;, 7-~) is also a (1, 2)- BrS. Conversely, if (Y, ~-;, ~-~) is a (1, 2)- BrS and Y E 2-Z)(X), then every set A c 1-$Z)(X)c~ 2-65(X) intersects Y. Proof. First, suppose that in our conditions, there exists a BsS (Y, ~-;, ~-~) of X such that (Y, T{, ~-~) is not (1, 2)-BrS. Then, by virtue of the first equivalence in (4) of Theorem 4.1.6 with (2) of Theorem 4.1.4 taken into account, there is a countable o(3
family {U~}~~
of 1-dense 2-open subsets of (Y, T~, ~-~) such that ('1 U~ g 1-79(Y), n=l
and so there exists a set H c ~-; \ {2~} together with (N3
HA n n=l
OO
U~ - N (H N U~) - ~.
(1)
n=l
Let U~ - g N U~ 9 ~-~' in (H, T[', T~') for each n - 1, oc. Then T 1" c l V ~ -
T 1" c l ( H N V ~ ) -
T 1' c l ( H C ~ U ~ ) N H -
T 1' c l H N H - H
as U~ e 1-~P(Y) for each n = 1, oc. Thus, there exists a countable family {U~}~__I oo
of 1-dense 2-open subsets of the BsS (H,T;',T~') and by (1),
n u~ - 2~. It is n=l
obvious that Un E ~-~' and H c T1' implies U~ e ~-~ in (I<, 7-; , ~-~) for each n - 1 , oc. For every n c N let us consider V/ c ~-2, E c 7"1 such that V~ n Y = Un and EAY=H. IfVn=V~nE, then Vn N Y = V~ N E O Y C E N Y = H a n d s o V~ N Y = U~ for each n - 1, oc. Clearly, Vn is 2-open in (E, v-~", 7-(~") for each n - 1, oc. Let us show that V~ E 1-D(E) for each n = 1, oc. If there exists a set V~,,-~ 1-D(E), then there is a set r c T [ " \ {;~} such that T N V~,, - 2~. In view of E C T1, we have T E T1 \ {2~}. Hence Y c 1-D(X) gives T N Y = T' =/= 2~. But T' C E N Y = H so that T' n (V~,, O Y) = T' V/Un,, = 2~, which is impossible since T' E T [ ' \ {2~} in (H, ~-~',~-~') and Ur~o ~ 1-D(H). Thus V~ r 1-D(E) for each n = 1, oc. Following (1) of Corollary 4.1.7, (E, "r~", v-~") is also a (1, 2)- BrS and by the first equivalence in (x)
(4) of Theorem 4.1.6 in conjunction with (2) of Theorem 4.1.4, we obtain O V~ n=l
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142
IV. Baire-Like Properties of Bitopological Spaces
Therefore, we have: V--
V1 ~ V2 ~ . . . ~ Vn ~ . . . .
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n=l
n=l
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oo
and, hence, n T l c l V n r
oo
n(vaun)r
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rt=l
4.2. Almost-(2, 1)-Baire Spaces First, note that for a BS (X, rl < r2) the condition (2) of Theorem 4.1.4 can be made more precise as follows:
Theorem 4.2.1. A BS (X, rl < r2) is an A - ( 2 , 1 ) - B r S if and only if the intersection of any monotone decreasing sequence of 2-dense 1-open sets is 2-dense in X . Pro@ It suffices to prove t h a t if the intersection of any monotone decreasing sequence of 2-dense 1-open sets is 2-dense in X, then (3) of Theorem 4.1.4 also holds. Let A c ( 2 , 1 ) - C a t g i ( X ) be any set. Then, by analogy with the proof of the implication ( 2 ) ~ ( 3 ) in Theorem 4.1.4, we can assume t h a t A c 1 - ~ ( X ) N (x)
(2, 1 ) - d a t g i ( X ) so t h a t A -
U An where AN C coT1 n (2, 1)-N'D(X) for each n=l oo
n -
1, oc.
Suppose that X \ A ~ 2 - D ( X ) ,
that is,
n (X\AN)E2-D(X).
Let
n=l n
U~ -
n ( x \ Ak) for each n - 1, oc. It is obvious t h a t {U~}n~__l is a monotone k=l
decreasing sequence. Moreover, n
x\u
n
UA,
-x\ k=l
k=l
where Ak C COT 1 n (2,1)-AfD(X) for each k = 1, oc and for each n = 1, oc. According to Corollary 1.1.15, X \ g n C c o 7"1 n (2, 1)-N'D(X) (x)
and so U~ c T1 n 2 - ~ ) ( X ) for each n - 1, oc. Hence, by condition, n u~ E 2-79(x). rt=l
But oo
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4.2. Almost-(2, 1)-Baire Spaces
145
that (X \ A) \ B c 2-Z)(X \ A). Thus for every set B E (2, 1)-Catg I ( x \ A) the complement (X \ A) \ B c 2-/)(X \ A) and once more applying (3) of Theorem 4.1.4 gives that (X \ A, 7-~,7-~) is an A-(2, 1)- BrS. To prove the converse, suppose first that (X, T1, ~-2) contains no nonempty sets of (2,1)-first category. Then every set U c T2 \ {2~} is of (2,1)-second category in X and thus by (1) of Theorem 4.1.4, (X, 7-1,7-2) is an A-(2, 1)- BrS. If there exists (x)
a set A c (2, 1)- Catgi (X) \ {2~}, then A -
U AN, where ~-2intT1 clAN - 2~ for rt--1
each n - 1, oc and AN,, # ~ for some no C N. Clearly, An,, c (2,1)- Catgl(X) and following the condition, (X\A~,,, ~-~,7-~) is an A-(2, 1)- BrS. Since ~-2int A~,, c T2 int Tx cl An o - ~, we obtain X \ A~I, c 2-/)(X). Hence, it s u m c e s to use (~) of Proposition 4.2.2. K1 C o r o l l a r y 4.2.6. Let (X, T1 < 7-2) be an A-(2, 1)-BrS. Then A E 1-Gs(X)O 2-/)(X) implies that (A, ~-~,7-~) is an A-(2, 1)-BrS.
Proof. Indeed, A c 1-~5(X) N 2-/)(X) implies that X \ A c 1-3cr and so X \ A c (2, 1)-Catgi(X ).
N
2-Bd(X)
V