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Topology Proceedings 38 Volume 38, 2011 Pages 149{164 http://topology.auburn.edu/tp/ C(휏)-Cosmic Spaces by D. N. Georgiou, S. D. Iliadis, and A. C. Megaritis Electronically published on August 26, 2010 Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 38 (2011) Pages 149-164 E-Published on August 26, 2010 C(휏)-COSMIC SPACES D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS Abstract. In this paper we introduce and study the notion of C(휏)-cosmic space, where 휏 is an infinite cardinal. Partic- ularly, we prove that in the class of all C(휏)-cosmic spaces of weight ≤ 휏, there exists a universal element. 1. Preliminaries In what follows, by ! and c, we denote the first infinite cardinal and the cardinality of the continuum, respectively. Also, by jXj, we denote the cardinality of a set X and by w(X), the weight of a space X. A space T is said to be universal (see [3]) in a class IP of spaces if (1) T 2 IP and (2) for every X 2 IP there exists an embedding e of X into T . A regular space X is called cosmic (see [5]) if there exists a collection P of subsets of X with the properties (1) for every open subset U of X and every x 2 U, there exists P 2 P such that x 2 P ⊆ U, (2) jPj ≤ !. Recall that a family P of subsets of a space X is called a network of X (see [3]) if, for every point x 2 X and every open neighborhood 2000 Mathematics Subject Classification. Primary 54B99, 54C25, 54A05. Key words and phrases. cosmic space, C(휏)-cosmic space, saturated class, universal space. Work supported by the Caratheodory Programme of the University of Patras. ⃝c 2010 Topology Proceedings. 149 150 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS Ux of x, there exists P 2 P such that x 2 P ⊆ Ux. The network weight of a space X, denoted by nw(X), is defined as the least cardinal number 휏 such that X has a network of cardinality ≤ 휏. We note that a space X is cosmic if and only if it is regular and nw(X) ≤ !. Let 휏 be an infinite cardinal. A space X is called 휏-monolithic (see [1] and [2]) if, for every subset A of X with jAj ≤ 휏, we have nw(Cl(A)) ≤ 휏. In section 2, we give the notion of C(휏)-cosmic space and basic properties for this notion. In section 3, we prove that in the class of all C(휏)-cosmic spaces of weight ≤ 휏 there exists universal element. Finally, in section 4, we give some open problems. 2. Basic properties Definition 2.1. Let 휏 be an infinite cardinal. A space X is called Closed(휏)-cosmic (C(휏)-cosmic) if there exists a collection K of closed subsets of X with the properties (1) for every open subset U of X and every x 2 U, there exists K 2 K such that x 2 K ⊆ U, and (2) jKj ≤ 휏. Remark 2.2. (1) It is clear that for every C(휏)-cosmic space X we have nw(X) ≤ 휏. Particularly, if X is Hausdorff and compact, then we have w(X) = nw(X) ≤ 휏 (see [3, Theorem 3.1.19]). (2) Every C(휏)-cosmic space is C(휏1)-cosmic space for every 휏1 ≥ 휏. (3) Every regular space of weight less than or equal to 휏 is C(휏)- cosmic. (4) Every regular C(!)-cosmic space is cosmic. (5) Every C(휏)-cosmic is 휏-monolithic. (6) Every T1-space X with jXj ≤ 휏 is C(휏)-cosmic. 휏 (7) Every T1-space X with w(X) ≤ 휏 is C(2 )-cosmic. Indeed, for 휏 every T1-space X with w(X) ≤ 휏, we have that jXj ≤ 2 . Thus, by (6) the space X is C(2휏 )-cosmic. (8) Let X be the set of real numbers with the topology t = f(훼; +1): 훼 2 Xg [ f;g. For the space (X; t), we have that nw(X) ≤ !, X is not regular, and X is not C(휏)-cosmic for every infinite cardinal 휏. (9) Let X = [−1; 1] and t = fU ⊆ X : 0 2= U or (−1; 1) ⊆ Ug. Obviously, t is a topology on X. The space X is T0 and T4, but C(휏)-COSMIC SPACES 151 neither T1 nor T2. The space (X; t) is not C(휏)-cosmic for every infinite cardinal 휏. (10) Let X be a discrete space with jXj = 휏, where 휏 is an infinite cardinal. The space X is C(휏)-cosmic. However, X is not C(휈)- cosmic for every infinite cardinal 휈 < 휏. (11) Consider a dense subspace D of cardinality 휏 of the Cantor cube 2휏 . Then D is a C(휏)-cosmic regular space. Also, it has character 2휏 , and hence w(D) > 휏. (12) Consider the space X = (! × !) [ f1g in which a subset U is open if and only if U is empty or 1 2 U and U n (fng × !) is finite for all n 2 !. Then X is T0, cosmic (being countable), has uncountable weight (the neighborhoods of 1 are the same as in the countable Fr´echet fan), and has no closed network (because the singleton f1g is dense in X). (13) If, in Definition 2.1, the elements K of the family K are regular closed sets (that is, Cl(Int(K)) = K), then the class of all C(휏)- cosmic spaces is exactly the class of all regular spaces of weight less than or equal to 휏. (14) Let X be a non-compact locally compact space of weight less than or equal to 휏. Then the Alexandroff compactification !X of X is a C(휏)-cosmic space. In what follows we give special examples of C(휏)-cosmic spaces. In particular, we give Hausdorff C(휏)-cosmic spaces which are not regular. Example 2.3. (1) Let X be the set of real numbers with the topology t = fU ⊆ X : jX n Uj ≤ !g [ f;g: The space (X; t) is T1-space of weight c and it is not regular. Also, setting K = ffxg : x 2 Xg, we take a family of closed subsets of X such that jKj = c and for every open subset U of X and every x 2 U, we have x 2 fxg ⊆ U. This means that the space (X; t) is C(c)-cosmic. (2) Let P = f(훼; 훽) 2 R2 : 훽 > 0g be the open upper half-plane with the Euclidean topology T and L = f(훼; 훽) 2 R2 : 훽 = 0g. We consider the set X = P [ L with the topology t = T [ ffxg [ (P \ Ux): x 2 L; Ux 2 T; and x 2 Uxg. The space (X; t) is Hausdorff and it is not regular. Moreover, since 152 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS the subspace L of X is discrete, the weight of X is c. Setting K = ffxg : x 2 Xg, we take a family of closed subsets of X such that jKj = c and for every open subset U of X and every x 2 U, we have x 2 fxg ⊆ U. Therefore, the space (X; t) is C(c)-cosmic. (3) Let X be the set of real numbers with open neighborhoods of every nonzero point being as in the usual topology, while open 1 neighborhoods of 0 will have the form U n f n : n = 1; 2;:::g, where U is an open neighborhood of 0 in the usual topology. The space X is Hausdorff of weight !, is C(!)-cosmic, and is not regular. Proposition 2.4. Every subspace of a C(휏)-cosmic space is a C(휏)- cosmic space. Proof: Let X be a C(휏)-cosmic space and A ⊆ X. Then there exists a collection K of closed subsets of X with the properties (1) for every open subset U of X and every x 2 U, there exists K 2 K such that x 2 K ⊆ U, and (2) jKj ≤ 휏. We consider the collection KA ≡ fA \ K : K 2 Kg of closed subsets of A. The family KA satisfies the properties (3) for every open subset U A of A and every x 2 U A, there A A A exists K 2 KA such that x 2 K ⊆ U , and (4) jKAj ≤ 휏. The fourth property is clear. For the third, we consider an open subset U A of A and an element x of U A. Then there exists an open neighborhood U of x in X such that U A = A \ U. By the first property, there exists an element K of K such that x 2 K ⊆ U. A A A A Setting K = A \ K, we have K 2 KA and x 2 K ⊆ U . □ Q Proposition 2.5. The product 휆2Λ X휆 of a family fX휆 : 휆 2 Λg of C(휏)-cosmic spaces, where jΛj ≤ 휏, is a C(휏)-cosmic space. Proof: For every family of C(휏)-cosmic spaces X휆, 휆 2 Λ, there exists a collection K휆 of closed subsets of X휆 with the properties (1) for every open subset U휆 of X휆 and every x 2 U휆, there exists K휆 2 K휆 such that x 2 K휆 ⊆ U휆, and (2) jK j ≤ 휏.
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