ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET

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1 Novi Sad J. Math. Vol. 40, No. 2, 2010, 7-16 ON COUNTABLE FAMILIES OF TOPOLOGIES ON A SET M.K. Bose 1, Ajoy Mukharjee 2 Abstract Considering a countable number of topologies on a set X, we introduce the notion of (ℵ 0 )topological spaces as a generalization of the notions of both bitopological spaces and (ω)topological spaces, and study some of their properties. AMS Mathematics Subject Classification (2000): 54A10 Key words and phrases: bitopological space, (ω)topological space, (ℵ 0 )topological space, (ℵ 0 )open set, (m)hausdorff, (ℵ 0 )Hausdorff, (ℵ 0)regular, (ℵ 0)normal, completely (ℵ 0)normal, (ℵ 0)compact, (ℵ 0 )paracompact, (β-ℵ 0 )paracompact 1. Introduction The purpose of this paper is to study a space equipped with countable number of topologies. The notion of bitopological spaces was introduced in Kelly [5] and the notion of (ω)topological spaces in Bose and Tiwari [1]. By generalizing both of these notions, we introduce here the notion of (ℵ 0 )topological spaces (Definition 2.1). In the (ℵ 0 )topological spaces, we define (ℵ 0 )Hausdorffness, (ℵ 0 )regularity, (ℵ 0 )normality and (ℵ 0 )compactness. We also define complete (ℵ 0 )regularity and complete (ℵ 0 )normality. We prove some results for these notions. A set X equipped with an increasing sequence {T n } of topologies is called a (ω)topological space. If T n = T n for all n, n, then the (ω)topological space X becomes a topological space. Thus the notion of (ω)topological spaces generalizes the notion of topological spaces. But it does not generalize the notion of bitopological spaces. Fletcher et al. [4] attempted to define pairwise paracompactness in a bitopological space. But in the presence of pairwise Hausdorffness, the two topologies coincide and the resulting single topological space is paracompact whenever the bitopological space is pairwise paracompact. Later Datta [2] introduced a notion of pairwise paracompactness. In the (ℵ 0 )topological spaces, we introduce (ℵ 0 )paracompactness. A space which is (ℵ 0 )Hausdorff and (ℵ 0 )paracompact is 1 Department of Mathematics, University of North Bengal, Siliguri, W. Bengal , India, manojkumarbose@yahoo.com 2 Department of Mathematics, St. Joseph s College, North Point, Darjeeling, W. Bengal , India, ajoyjee@yahoo.com

2 8 M.K. Bose, A. Mukharjee (ℵ 0 )regular. Raghavan and Reilly [8] introduced α-, β-, γ- and δ-pairwise paracompactness. They presented a δ-pairwise paracompactness version of Michael s characterization (Michael [7]) of regular paracompact spaces. Unfortunately, the proof of this result is not correct (Kovár [6]). We introduce here a (β- ℵ 0 )paracompactness and prove the Michael s theorem for (β-ℵ 0 )paracompactness of the space X when it is (ℵ 0 )regular (Theorem 3.16). From this, we obtain an analogue of Michael s theorem for the β-pairwise paracompactness of a pairwise regular bitopological space, as a particular case. 2. Definitions Let {P n } be a sequence of topologies on a set X. The sequence {P n } is said to satisfy the condition ( ) (resp. condition (a)) if for any positive integer m, the union (resp. intersection) of a finite number of sets P n is a set P n. We introduce the following definitions. Definition 2.1. If {P n } is a sequence of topologies on a set X satisfying the condition ( ), then the pair (X, {P n }) is called an (ℵ 0 )topological space. A set G n P n is called an (ℵ 0 )open set. Throughout the paper, R and N denote the set of real numbers and the set of positive integers respectively. Members of N are generally denoted by k, l, m, n etc. If T is a topology on a set X, then (T )cla and (T )inta denote the closure and interior respectively of A X with respect to T. Unless otherwise mentioned, X denotes the (ℵ 0 )topological space (X, {P n }). Elements of X are denoted by x, y etc. All the sets considered here are subsets of X. Example 2.1. We enumerate all the rational numbers: r 1, r 2, r 3,.... Let Q n denote the particular point topology (Steen and Seebach, Jr. [9]) on R, where each nonempty set Q n contains r n. Then (R, {Q n }) is an (ℵ 0 )topological space, where the sequence {Q n } does not satisfy the condition (a). In the (ℵ 0 )topological space considered in Example 2.2, {P n } satisfies the condition (a). Definition 2.2. Suppose Y X and P n Y denotes the subspace topology on Y induced by P n. Then (Y, {P n Y }) is called a subspace of (X, {P n }). Definition 2.3. A function f : X R is said to be ( P n )upper semicontinuous (resp. ( P n )lower semi-continuous) if for every a R, f 1 ((, a)) P n (resp. f 1 ((a, )) P n ) where M N. Definition 2.4. X is said to be (ℵ 0 )compact if every (ℵ 0 )open cover C of X with C P n for at least two values of n has a finite subcover. Let M N. X is said to be (M)compact if every cover of X consisting of the sets has a finite subcover. P n

3 On countable families of topologies on a set 9 Definition 2.5. An (ℵ 0 )open cover {U α U α P nα } of X is said to be shrinkable if there is another (ℵ 0 )open cover {V α V α P nα } of X with (P n )clv α U α for some n n α. Definition 2.6. A set D is called a (P n )neighbourhood, or in short a (P n )nbd, of a set A if A (P n )intd. Definition 2.7. X is said to be (m)hausdorff if for any pair of distinct points x, y X, there exist U P m and V P n such that x U, y V and U V =. If X is (m)hausdorff for all m N, then X is said to be an (ℵ 0 )Hausdorff space. The (ℵ 0 )topological space (R, {Q n }) considered in Example 2.1, is not (n)hausdorff for any n. Also for any n, the topological space (R, Q n ) is not Hausdorff and for any pair of integers m, n, the bitopological space (R, Q m, Q n ) is not pairwise Hausdorff. Example 2.2. Let the sequence {P n } of topologies on R be defined by P 1 = U, P n = { } {G (n, ) G U} for all n > 1, where U is the usual topology on R. Then the (ℵ 0 )topological space (R, {P n }) is (1)Hausdorff but it is not (n)hausdorff for any n 1, and so it is not (ℵ 0 )Hausdorff. Definition 2.8. X is said to be (m)regular if for any point x X and any (P n )closed set A with n m and x / A, there exist U P n and V P m such that x U, A V, U V =. X is said to be (ℵ 0 )regular if it is (m)regular for all m N. It is easy to see that X is (m)regular iff for any point x X and any (P n )open set G containing x with n m, there exists a U P n containing x such that (P m )clu G. Definition 2.9. X is said to be completely (m)regular if for any point x X and any (P n )closed set A with n m and x / A, there exists a function f : X [0, 1] such that f(x) = 0, f(y) = 1 for all y A, f is ( P n )upper semi-continuous and (P m )lower semi-continuous. X is completely (ℵ 0 )regular if it is completely (m)regular for all m N. Definition X is said to be (m)normal if given a (P m )closed set A and a (P n )closed set B with n m and A B =, there exist U P n and V P m such that A U, B V and U V =. If X is (m)normal for all m N, then it is said to be (ℵ 0 )normal.

4 10 M.K. Bose, A. Mukharjee From the definition, it follows that X is (m)normal iff for any (P m )closed set A and any (P n )open set W containing A with n m, there exists a U such that A U (P m )clu W. P n Definition X is said to be completely (m)normal if for each pair A, B of subsets of X satisfying (A (P n )clb) ((P m )cla B) =, n m, there exist U P n and V P m such that A U, B V and U V =. Complete (ℵ 0 )normality is defined in the obvious way. The (ℵ 0 )topological space (R, {Q n }) considered in Example 2.1, is completely (ℵ 0 )normal but for no n, the topological space (R, Q n ) is normal. Also, for no n, (R, {Q n }) is (n)regular. Example 2.3. Let X be an infinite set and a, b, c X. Suppose G (resp. H) is a collection of subsets of X, which contains, in addition to and X, all those subsets E of X for which a / E and c E (resp. a, b / E and c E). Then G and H form topologies on X. Let P n = G for odd n, and = H for even n. Then the (ℵ 0 )topological space (X, {P n }) is (ℵ 0 )normal but not completely (ℵ 0 )normal. In fact, it is completely (m)normal for no m. Definition A collection C of subsets of X is said to be (ℵ 0 )locally finite if each x X has a (P n )open nbd, for at least two values of n, intersecting at most finitely many U C. For any m N, C is said to be ( P n )locally finite if for each x X, there exists a set P n, containing x and intersecting at most finitely many U C. Definition A refinement V of an (ℵ 0 )open cover U is said to be a parallel refinement if V V is (P n )open whenever V U U and U is (P n )open. Definition X is said to be (ℵ 0 )paracompact if every (ℵ 0 )open cover U of X with U P n for at least two values of n has an (ℵ 0 )locally finite parallel refinement. It follows from the definitions that an (ℵ 0 )compact space is (ℵ 0 )paracompact. The converse is not true: Example 2.4. Let T be the indiscrete topology on R, and U n be the subspace topology U I n on I n = ( n, n), of the usual topology U on R. If P n = T U n, then (R, {P n }) is an (ℵ 0 )topological space, which is (ℵ 0 )paracompact but not (ℵ 0 )compact. Let P denote the smallest topology containing all the topologies P n, n N. We call a cover C of X, a ( P n )cover if C P n. Definition X is said to be (β-ℵ 0 )paracompact if for all m N, every ( P n )cover of X has a ( P n )locally finite (P)open refinement.

5 On countable families of topologies on a set 11 The following example shows that a (β-ℵ 0 )paracompact space may not be (ℵ 0 )paracompact. Example 2.5. Suppose X = [0, ) and U is the usual topology on X. Let P 1 = discrete topology on X, P 2 = U and for n > 2, P n = { } {G (n, ) G U}. Then the (ℵ 0 )topological space (X, {P n }) is (β-ℵ 0 )paracompact but not (ℵ 0 )paracompact. Open question. Does (ℵ 0 )paracompactness of a space imply (β-ℵ 0 )paracompactness of the space? 3. Results Theorem 3.1. If X is (ℵ 0 )compact, and K is (P m )closed for some m, then K is (ℵ 0 )compact. If K is a proper subset of X, then K is (P n )compact for all n m. Proof. We prove only the second part. Choose n 0 m. Let U be a (P n0 )open cover of K. Then U 1 = U {X K} is an (ℵ 0 )open cover of X with U 1 P n for two values of n. Therefore U 1 has a finite subcover. Hence U has a finite subcover. Theorem 3.2. If (X, P n ) is a Hausdorff topological space for each n, and (X, {P n }) is (ℵ 0 )compact, then all P n are identical. The proof is straightforward. Theorem 3.3. Let M = {n N n m}. If X is (m)hausdorff and (M)compact, then P n P m for all n. Proof. Let n m. If P n P m, then there exists a set U P n such that U / P m. Then X U is (P n )closed, and hence (M)compact. Since X U is not (P m )closed, there is a point p U which is a (P m )limit point of X U. Again, since X is (m)hausdorff, for each x X U, there exist U x U P n and V x P m such that x U x, p V x and U x V x =. Then {U x x X U} is a cover of X U by sets P n. Therefore it has a finite subcover U x1, U x2,..., U xn. If V = n V xi, then V P m, p V and V (X U) =, which is a contradiction, since p is a (P m )limit point of X U. Theorem 3.4. X is (m)hausdorff iff for all x, {x} = {(P n )clu n m, U P m, x U}. The proof is straightforward. Theorem 3.5. If X is (m)hausdorff, then for each filterbase F on X such that F is (P m )convergent to x and (P n )convergent to y for some n m, x = y. The converse is also true if {P n } satisfies the condition (a).

6 12 M.K. Bose, A. Mukharjee Proof. Let X be (m)hausdorff, and F be a filterbase on X, which is (P m )convergent to x. Let y x. Then there exist U P m and V P n such that x U, y V, U V =. There is an A F with A U. If V P n, n m, then F cannot be (P n )convergent to y. For if it is so, there must be a B F with B V, which is not possible, since any two elements of F have a non-empty intersection. To prove the converse, suppose X is not (m)hausdorff. Then there exists a pair x, y (x y) such that for any U P m and V P n with x U, y V, we have U V. So the class C = {U V U P m, V P n, x U, y V } is a filterbase on X, which is (P m )convergent to x and (P n )convergent to y for all n m. If X is (ℵ 0 )Hausdorff, then Theorem 3.5 is true for any pair of elements m, n N. Likewise, each of the following theorems with (m) has a corresponding result with (ℵ 0 ). Theorem 3.6. If X is (m)hausdorff and M = {n n m}, then (M)compact subsets are (P m )closed. Proof. Let K be an (M)compact subset of X and x X K. If y K, then there exist sets U xy P m and V xy P n such that x U xy, y V xy and U xy V xy =. Then V = {V xy y K} forms a cover of K by sets P n. Therefore V has a finite subcover V xy1, V xy2,..., V xyk. If V x = k U x = k U xyi, then x U x P m, K V x X K {U x x X K} {X V x x X K} X K. X K = {U x x X K}. V xyi P n and U x V x =. Now and Thus K is (P m )closed. Theorem 3.7. X is (m)regular iff for n 0 m, any (P n0 )closed set A is the intersection of all (P n )closed (P m )nbds of A, n m. Proof. Let X be (m)regular. Suppose x belongs to the complement X A of the (P n0 )closed set A with n 0 m. Since X A is (P n0 )open, n 0 m, there exists a set G P n such that x G (P m )clg X A. Then A X (P m )clg X G. Thus X G is a (P n )closed (P m )nbd of A with

7 On countable families of topologies on a set 13 n m and x / X G. Hence the intersection of all (P n )closed (P m )nbds of A with n m, is the set A. To prove the converse, suppose for n 0 m, A is a (P n0 )closed set not containing the point x. Then there exists a (P n )closed (P m )nbd F of A with n m and x / F. If U = X F and V = (P m )intf, then U P n, V P m, x U, A V and U V =. Theorem 3.8. If X is (ℵ 0 )compact and (m)hausdorff, and if {P n } satisfies the condition (a), then X is (m)regular. Proof. Let x X and A be a (P n )closed set with n m and x / A. If y A, then by (m)hausdorffness of X, there exist U y P n and V y P m such that x U y, y V y and U y V y =. Then V = {V y y A} forms a (P m )open cover of A. Since A is (P n )closed, by Theorem 3.1, it is (P m )compact, and hence V has a finite subcover V y1,..., V yk. If U = k U yi and V = k V yi, then U P n, V P m, x U, A V and U V =. Theorem 3.9. Every completely (m)regular space is (m)regular. The proof is straightforward. Theorem If X is (ℵ 0 )compact and (m)regular, then X is (m)normal. Proof. Suppose A is a (P m )closed set and B is a (P n )closed set with n m and A B =. For x A, there is a U x P n and V x P m such that x U x, B V x, U x V x =. Since U = {U x x A} is a cover of A with sets P n, and X is (ℵ 0 )compact, it follows that there is a finite subcollection U x1, U x2,..., U xk of U covering A. If U = k U xi and V = k V xi, then U P n, V P m, U V =, A U, B V. Corollary 3.1. If X is (ℵ 0 )compact and (m)hausdorff, and if the condition (a), then X is (m)normal. {P n } satisfies The following theorem is a generalization of Urysohn s Lemma for (ℵ 0 )topological spaces. Theorem Suppose X is (m)normal. Let A be a (P m )closed set and B be a (P n )closed set with n m and A B =. Then there exists a function f : X [0, 1] such that f(x) = 0 for all x A, f(x) = 1 for all x B, f is ( P n )upper semi-continuous and (P m )lower semi-continuous. The proof is omitted. It is similar to the proof of Theorem 2.7 (Kelly [5]). It is easy to see the converse of the above theorem is also true.

8 14 M.K. Bose, A. Mukharjee From the definitions, it is clear that a completely (m)normal space is (m)normal. Also, it can be shown that complete (m)normality is a hereditary property but (m)normality is not hereditary. The (m)hausdorff property, (m)regularity and complete (m)regularity are hereditary. Theorem X is completely (m)normal iff every subspace of X is (m)normal. The proof is omitted. Theorem If X is (ℵ 0 )normal, then every finite (ℵ 0 )open cover U = {U 1, U 2,..., U k } of X with U i P ni and n i n i if i i, is shrinkable. Proof. Suppose X is (ℵ 0 )normal. We choose a finite (ℵ 0 )open cover U = {U 1, U 2,..., U k } of X, satisfying the above conditions. Then (X U 1 ) (X U 2 )... (X U k ) =, and so (X U 1 ) and (X U 2 )... (X U k ) are disjoint. Again, X U 1 is (P n1 )closed and (X U 2 )... (X U k ) is (P n0 )closed for some n 0 n 1. So there exist V 1 P n for some n n 1 and W 1 P n1 such that X U 1 V 1, (X U 2 )... (X U k ) W 1 and V 1 W 1 =. Then, (P n )clw 1 X V 1 U 1, n n 1. Therefore it follows that {W 1, U 2,..., U k } forms an (ℵ 0 )open cover of X with (P n )clw 1 U 1, n n 1. If we apply the process k times, we obtain an (ℵ 0 )open cover {W 1, W 2,..., W k } such that W i P ni and (P n )clw i U i, n n i. Theorem If X is (ℵ 0 )paracompact, and if K is a (P m )closed subset of X for some m, then K is (ℵ 0 )paracompact. The proof is straightforward. Theorem If X is (m)hausdorff and (ℵ 0 )paracompact, and if {P n } satisfies the condition (a), then X is (m)regular. The proof is omitted. Theorem Let X be (ℵ 0 )regular, and let {P n } satisfy the condition (a). Then X is (β-ℵ 0 )paracompact iff for all m N, every ( P n )cover of X has k=1 a (P)open refinement V = V k, where each V k is ( P n )locally finite. Proof. Since the only if part of the theorem is obviously true, we prove the if part. It is done in three steps. Step I. For m N, let G be a ( P n )cover of X. Then it has a (P)open k=1 refinement V = V k, where each V k is ( P n )locally finite. Suppose V k = {V kα α A}. Let V kα G kα G. We write W k = G kα, W i = W k, A 1 = α k=1 W 1, and for i > 1, A i = W i W i 1. Then {W k k N} is a cover of X, and i

9 On countable families of topologies on a set 15 so {A i i N} is a cover of X. This cover is ( P n )locally finite. fact, if k(x) is the first k for which x W k(x), then x G k(x)α for some α, G k(x)α P n, and G k(x)α does not intersect any A i for i > k(x). Thus E(G) = {E kα k N, α A}, where E kα = A k V kα, is a ( P n )locally finite refinement of G. Step II. For m N, let H be a ( In P n )cover of X. For x X, we choose an H x H such that x H x. Since X is (ℵ 0 )regular, and since H x is (P n )open with n m, there exists a set Hx 1 P n such that x Hx 1 (P m )clhx 1 H x. Since P m P, we have (1) (P)clH 1 x (P m )clh 1 x H x. Now H 1 = {H 1 x x X} is also a ( get a ( P n )cover of X. Hence by Step I, we P n )locally finite refinement E(H 1 ) of H 1. For E E(H 1 ), we get an Hx 1 H 1 with E Hx 1 and hence by (1), (P)clE H x. Thus F(H) = {(P)clE E E(H 1 )} is a (P)closed refinement of H. Also F(H) is ( P n )locally finite: Since E(H 1 ) is ( P n )locally finite, there exists, for x X, a G P n, n m such that x G, and for all but finitely many E E(H 1 ), G E = G (P n )cle = G (P)clE =. Step III. We choose an m N. Let U be a ( P n )cover of X. By Step I, U has a ( P n )locally finite refinement E(U). For x X, suppose W x P n is a set containing x and intersecting a finite number of members of E(U). Then W = {W x x X} forms a ( P n )cover of X. Therefore by Step II, W has a ( P n )locally finite (P)closed refinement F(W). For E E(U), let S E = X {F F(W) F E = }. Since the collection F(W) of (P)closed sets is ( P n )locally finite, and since P n P, by 9.2 (Dugundji [3], p. 82), it follows that the set {F F(W) F E = } is (P)closed, and hence S E is (P)open. Since E(U) is a cover of X, S = {S E E E(U)} is a (P)open cover of X. S is also ( P n )locally finite. For, consider a set D x P n containing x and intersecting F 1, F 2,..., F k

10 16 M.K. Bose, A. Mukharjee F(W). Now D x S E, F i S E for some i = 1, 2,..., k, F i E for some i = 1, 2,..., k. Since each F i is contained in some W x, it can intersect at most finitely many E E(U). Therefore D x can intersect at most finitely many S E S. For every E E(U), we take U E U such that E U E. Then the collection {U E S E E E(U)} is a ( P n )locally finite (P)open refinement of U. Therefore X is (β-ℵ 0 )paracompact. Acknowledgement. We are grateful to the referee for careful reading the paper and for certain suggestions towards the improvement of the paper. References [1] Bose, M.K., Tiwari, R., On increasing sequences of topologies on a set. Riv. Mat. Univ. Parma (7) 7 (2007), [2] Datta, M.C., Paracompactness in bitopological spaces and an application to quasi-metric spaces. Indian J. Pure Appl. Math. 8 (1977), [3] Dugundji, J., Topology. Boston: Allyn and Bacon [4] Fletcher, P., Hoyle III, H.B., Patty, C.W., The comparison of topologies. Duke Math. J. 36 (1969), [5] Kelly, J.C., Bitopological spaces. Proc. Lond. Math. Soc. (3) 13 (1963), [6] Kovár, M.M., On 3-topological version of θ-regularity. Int. J. Math. Math. Sci. 23 (2000), [7] Michael, E., A note on paracompact spaces. Proc. Amer. Math. Soc. 4 (1953), [8] Raghavan, T.G., Reilly, I.L., A new bitopological paracompactness. J. Aust. Math. Soc. (Series A) 41 (1986), [9] Steen, L.A., Seebach Jr., J.A., Counterexamples in topology. New York: Holt, Rinehart and Winston Inc Received by the editors April 12, 2008