Volume 38, 2011 Pages 149–164

http://topology.auburn.edu/tp/

퐶(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

퐶(휏)-COSMIC SPACES

D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

Abstract. In this paper we introduce and study the notion of 퐶(휏)-cosmic space, where 휏 is an infinite cardinal. Partic- ularly, we prove that in the class of all 퐶(휏)-cosmic spaces of weight ≤ 휏, there exists a universal element.

1. Preliminaries In what follows, by 휔 and c, we denote the first infinite cardinal and the cardinality of the continuum, respectively. Also, by ∣푋∣, we denote the cardinality of a set 푋 and by 푤(푋), the weight of a space 푋. A space 푇 is said to be universal (see [3]) in a class IP of spaces if (1) 푇 ∈ IP and (2) for every 푋 ∈ IP there exists an embedding 푒 of 푋 into 푇 . A regular space 푋 is called cosmic (see [5]) if there exists a collection 풫 of subsets of 푋 with the properties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 푃 ∈ 풫 such that 푥 ∈ 푃 ⊆ 푈, (2) ∣풫∣ ≤ 휔. Recall that a family 풫 of subsets of a space 푋 is called a network of 푋 (see [3]) if, for every point 푥 ∈ 푋 and every open neighborhood

2000 Mathematics Subject Classification. Primary 54B99, 54C25, 54A05. Key words and phrases. cosmic space, 퐶(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

푈푥 of 푥, there exists 푃 ∈ 풫 such that 푥 ∈ 푃 ⊆ 푈푥. The network weight of a space 푋, denoted by 푛푤(푋), is defined as the least cardinal number 휏 such that 푋 has a network of cardinality ≤ 휏. We note that a space 푋 is cosmic if and only if it is regular and 푛푤(푋) ≤ 휔. Let 휏 be an infinite cardinal. A space 푋 is called 휏-monolithic (see [1] and [2]) if, for every subset 퐴 of 푋 with ∣퐴∣ ≤ 휏, we have 푛푤(Cl(퐴)) ≤ 휏. In section 2, we give the notion of 퐶(휏)-cosmic space and basic properties for this notion. In section 3, we prove that in the class of all 퐶(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 푋 is called Closed(휏)-cosmic (퐶(휏)-cosmic) if there exists a collection 풦 of closed subsets of 푋 with the properties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏. Remark 2.2. (1) It is clear that for every 퐶(휏)-cosmic space 푋 we have 푛푤(푋) ≤ 휏. Particularly, if 푋 is Hausdorff and compact, then we have 푤(푋) = 푛푤(푋) ≤ 휏 (see [3, Theorem 3.1.19]). (2) Every 퐶(휏)-cosmic space is 퐶(휏1)-cosmic space for every 휏1 ≥ 휏. (3) Every regular space of weight less than or equal to 휏 is 퐶(휏)- cosmic. (4) Every regular 퐶(휔)-cosmic space is cosmic. (5) Every 퐶(휏)-cosmic is 휏-monolithic. (6) Every T1-space 푋 with ∣푋∣ ≤ 휏 is 퐶(휏)-cosmic. 휏 (7) Every T1-space 푋 with 푤(푋) ≤ 휏 is 퐶(2 )-cosmic. Indeed, for 휏 every T1-space 푋 with 푤(푋) ≤ 휏, we have that ∣푋∣ ≤ 2 . Thus, by (6) the space 푋 is 퐶(2휏 )-cosmic. (8) Let 푋 be the set of real numbers with the topology 푡 = {(훼, +∞): 훼 ∈ 푋} ∪ {∅}. For the space (푋, 푡), we have that 푛푤(푋) ≤ 휔, 푋 is not regular, and 푋 is not 퐶(휏)-cosmic for every infinite cardinal 휏. (9) Let 푋 = [−1, 1] and 푡 = {푈 ⊆ 푋 : 0 ∈/ 푈 or (−1, 1) ⊆ 푈}. Obviously, 푡 is a topology on 푋. The space 푋 is T0 and T4, but 퐶(휏)-COSMIC SPACES 151 neither T1 nor T2. The space (푋, 푡) is not 퐶(휏)-cosmic for every infinite cardinal 휏. (10) Let 푋 be a with ∣푋∣ = 휏, where 휏 is an infinite cardinal. The space 푋 is 퐶(휏)-cosmic. However, 푋 is not 퐶(휈)- cosmic for every infinite cardinal 휈 < 휏. (11) Consider a dense subspace 퐷 of cardinality 휏 of the Cantor cube 2휏 . Then 퐷 is a 퐶(휏)-cosmic regular space. Also, it has character 2휏 , and hence 푤(퐷) > 휏. (12) Consider the space 푋 = (휔 × 휔) ∪ {∞} in which a subset 푈 is open if and only if 푈 is empty or ∞ ∈ 푈 and 푈 ∖ ({푛} × 휔) is finite for all 푛 ∈ 휔. Then 푋 is T0, cosmic (being countable), has uncountable weight (the neighborhoods of ∞ are the same as in the countable Fr´echet fan), and has no closed network (because the singleton {∞} is dense in X). (13) If, in Definition 2.1, the elements 퐾 of the family 풦 are regular closed sets (that is, Cl(Int(K)) = K), then the class of all 퐶(휏)- cosmic spaces is exactly the class of all regular spaces of weight less than or equal to 휏. (14) Let 푋 be a non-compact locally compact space of weight less than or equal to 휏. Then the Alexandroff compactification 휔푋 of 푋 is a 퐶(휏)-cosmic space.

In what follows we give special examples of 퐶(휏)-cosmic spaces. In particular, we give Hausdorff 퐶(휏)-cosmic spaces which are not regular.

Example 2.3. (1) Let 푋 be the set of real numbers with the topology 푡 = {푈 ⊆ 푋 : ∣푋 ∖ 푈∣ ≤ 휔} ∪ {∅}.

The space (푋, 푡) is T1-space of weight c and it is not regular. Also, setting 풦 = {{푥} : 푥 ∈ 푋}, we take a family of closed subsets of 푋 such that ∣풦∣ = c and for every open subset 푈 of 푋 and every 푥 ∈ 푈, we have 푥 ∈ {푥} ⊆ 푈. This means that the space (푋, 푡) is 퐶(c)-cosmic. (2) Let 푃 = {(훼, 훽) ∈ ℝ2 : 훽 > 0} be the open upper half-plane with the Euclidean topology 푇 and 퐿 = {(훼, 훽) ∈ ℝ2 : 훽 = 0}. We consider the set 푋 = 푃 ∪ 퐿 with the topology 푡 = 푇 ∪ {{푥} ∪ (푃 ∩ 푈푥): 푥 ∈ 퐿, 푈푥 ∈ 푇, and 푥 ∈ 푈푥}. The space (푋, 푡) is Hausdorff and it is not regular. Moreover, since 152 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS the subspace 퐿 of 푋 is discrete, the weight of 푋 is c. Setting 풦 = {{푥} : 푥 ∈ 푋}, we take a family of closed subsets of 푋 such that ∣풦∣ = c and for every open subset 푈 of 푋 and every 푥 ∈ 푈, we have 푥 ∈ {푥} ⊆ 푈. Therefore, the space (푋, 푡) is 퐶(c)-cosmic. (3) Let 푋 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 푈 ∖ { 푛 : 푛 = 1, 2,...}, where 푈 is an open neighborhood of 0 in the usual topology. The space 푋 is Hausdorff of weight 휔, is 퐶(휔)-cosmic, and is not regular. Proposition 2.4. Every subspace of a 퐶(휏)-cosmic space is a 퐶(휏)- cosmic space. Proof: Let 푋 be a 퐶(휏)-cosmic space and 퐴 ⊆ 푋. Then there exists a collection 풦 of closed subsets of 푋 with the properties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏.

We consider the collection 풦퐴 ≡ {퐴 ∩ 퐾 : 퐾 ∈ 풦} of closed subsets of 퐴. The family 풦퐴 satisfies the properties (3) for every open subset 푈 퐴 of 퐴 and every 푥 ∈ 푈 퐴, there 퐴 퐴 퐴 exists 퐾 ∈ 풦퐴 such that 푥 ∈ 퐾 ⊆ 푈 , and (4) ∣풦퐴∣ ≤ 휏. The fourth property is clear. For the third, we consider an open subset 푈 퐴 of 퐴 and an element 푥 of 푈 퐴. Then there exists an open neighborhood 푈 of 푥 in 푋 such that 푈 퐴 = 퐴 ∩ 푈. By the first property, there exists an element 퐾 of 풦 such that 푥 ∈ 퐾 ⊆ 푈. 퐴 퐴 퐴 퐴 Setting 퐾 = 퐴 ∩ 퐾, we have 퐾 ∈ 풦퐴 and 푥 ∈ 퐾 ⊆ 푈 . □ ∏ Proposition 2.5. The product 휆∈Λ 푋휆 of a family {푋휆 : 휆 ∈ Λ} of 퐶(휏)-cosmic spaces, where ∣Λ∣ ≤ 휏, is a 퐶(휏)-cosmic space.

Proof: For every family of 퐶(휏)-cosmic spaces 푋휆, 휆 ∈ Λ, there exists a collection 풦휆 of closed subsets of 푋휆 with the properties

(1) for every open subset 푈휆 of 푋휆 and every 푥 ∈ 푈휆, there exists 퐾휆 ∈ 풦휆 such that 푥 ∈ 퐾휆 ⊆ 푈휆, and (2) ∣풦 ∣ ≤ 휏. 휆 ∏ We consider the family 풦 of all closed subsets of 휆∈Λ 푋휆 of the ∩푛 −1 form 휋 (퐾 ), where 휆1, . . . , 휆푛 ∈ Λ, 휋 is the projection of 푖=1 휆푖 휆푖 휆푖 퐶(휏)-COSMIC SPACES 153 ∏ 푋 onto 푋 , and 퐾 ∈ 풦 , 푖 = 1, . . . , 푛. Obviously, the 휆∈Λ 휆 휆푖 휆푖 휆푖 family 풦 satisfies property (2) of Definition 2.1. We∏ prove property (1) of Definition 2.1. Let 푈 be an open subset of 휆∈Λ 푋휆 and 푥 ∈ 푈. Without loss of generality, we can suppose ∩푛 −1 that 푈 ≡ 휋 (푈 ), where 휆1, . . . , 휆푛 ∈ Λ and 푈 is open in 푖=1 휆푖 휆푖 휆푖 푋 , 푖 = 1, . . . , 푛. For 푖 = 1, . . . , 푛, there exists 퐾 ∈ 풦 such 휆푖 휆푖 휆푖 −1 −1 that 휋휆 (푥) ∈ 퐾휆 ⊆ 푈휆 and, therefore, 푥 ∈ 휋 (퐾휆 ) ⊆ 휋 (푈휆 ). 푖 ∩ 푖 푖 휆푖 푖 휆푖 푖 Setting 퐾 = 푛 휋−1(퐾 ), we have 퐾 ∈ 풦 and 푥 ∈ 퐾 ⊆ 푈. □ 푖=1 휆푖 휆푖 휇 Proposition 2.6. Let S = {푋휆, 푓휆 , Λ} be an inverse system where ∣Λ∣ ≤ 휏. If the spaces 푋휆, 휆 ∈ Λ, are 퐶(휏)-cosmic, then the inverse limit lim S is a 퐶(휏)-cosmic space. ←

Proof: Since the spaces∏ 푋휆, 휆 ∈ Λ, are 퐶(휏)-cosmic, by Propo- sition 2.5, the product 휆∈Λ 푋휆 is a 퐶(휏)-cosmic space. More- over, by Proposition 2.4, every subspace of a 퐶(휏)-cosmic∏ space is a 퐶(휏)-cosmic space. Therefore, the subspace lim S of 푋휆 is ← 휆∈Λ a 퐶(휏)-cosmic space. □ Proposition 2.7. Let 푓 be a continuous and closed map from a 퐶(휏)-cosmic space 푋 onto a space 푌 . Then 푌 is a 퐶(휏)-cosmic space. Proof: There exists a collection 풦 of closed subsets of 푋 with the properties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏. We consider the family 푓(풦) ≡ {푓(퐾): 퐾 ∈ 풦}. The family 풦 satisfies property (2) of Definition 2.1. Also, since 푓 is closed, every element of the family 푓(풦) is a closed subset of 푌 . So, for the family 풦, it suffices to prove property (1) of Definition 2.1. Let 푊 be an open subset of 푌 and 푦 ∈ 푊 . Since 푓 is onto, there exists 푥 ∈ 푋 such that 푓(푥) = 푦. Moreover, since the map 푓 is continuous, the subset 푓 −1(푊 ) of 푌 is an open neighborhood of 푥 in 푌 . There exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푓 −1(푊 ) and, therefore, 푦 ∈ 푓(퐾) ⊆ 푊 . For the subset 퐹 = 푓(퐾) ∈ 푓(풦), we have that 푦 ∈ 퐹 ⊆ 푊 . Thus, the space 푌 is a 퐶(휏)-cosmic space. □ 154 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

Corollary 2.8. The class of all 퐶(휏)-cosmic spaces is topological. That is, every space homeomorphic to a 퐶(휏)-cosmic space is a 퐶(휏)-cosmic. Definition 2.9. Let 퐶(푌, 푍) be the set of all continuous maps from a space 푌 to a space 푍. The pointwise topology on 퐶(푌, 푍) (see [3]) is the topology for which the family of all sets of the form (푦, 푈) = {푓 ∈ 퐶(푌, 푍): 푓(푦) ∈ 푈}, where 푦 ∈ 푌 and 푈 is open in 푍, is a subbasis. Proposition 2.10. Let 푌 be a space with ∣푌 ∣ ≤ 휏 and 푍 be a 퐶(휏)-cosmic space. The space 퐶(푌, 푍) with the pointwise topology is a 퐶(휏)-cosmic space.

Proof: The pointwise topology∏ on 퐶(푌, 푍) coincides with the topology of a subspace of 푦∈푌 푍푦, where 푍푦 = 푍 for every 푦 ∈ 푌 (see [3, Proposition 2.6.3 ]). Since 푍 is∏ a 퐶(휏)-cosmic space and ∣푌 ∣ ≤ 휏, by Proposition 2.5, the product 푦∈푌 푍푦 is a 퐶(휏)-cosmic space. By Proposition 2.4, the space 퐶(푌, 푍) is a 퐶(휏)-cosmic space. □ Remark 2.11. Let 푌 = 푍 = ℝ. Then ∣푌 ∣ = c > 휔, 푍 is cosmic, and the space 퐶푝(푌, 푍) is Tychonoff and cosmic. Now, let 푌 be a discrete space with ∣푌 ∣ = c and 푍 = ℝ. Then 푍 is cosmic and 푌 퐶푝(푌, 푍) = ℝ has the network weight c and, therefore, it is not cosmic. Proposition 2.12. Every 퐶(휔)-cosmic space 푋 is a perfect space; that is, each closed subset of 푋 is a 퐺훿-set. Proof: There exists a collection 풦 of closed subsets of 푋 with the properties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휔. Let 퐹 be an arbitrary closed subset of 푋. We prove that 퐹 is a 퐺훿-set. It suffices to show that 푋 ∖ 퐹 is an 퐹휎-set. For every 푥 ∈ 푋 ∖ 퐹 , there exists∪ 퐾푥 ∈ 풦 such that 푥 ∈ 퐾푥 ⊆ 푋 ∖ 퐹 . Therefore, 푋 ∖ 퐹 = 퐾 . Since ∣풦∣ ≤ 휔, there exist 푥 ∈ 푋 푥∈푋 푥 ∪ 푖 and 푖 ∈ 휔 such that 푋 ∖ 퐹 = 퐾 . Hence, 푋 ∖ 퐹 is an 퐹 - 푖∈휔 푥푖 휎 set. □ 퐶(휏)-COSMIC SPACES 155

Corollary 2.13 ([3]). Every regular space with a countable base is a perfect space. Proof: Follows immediately by the fact that every regular space with a countable base is 퐶(휔)-cosmic. □ Proposition 2.12 can be generalized as follows. Proposition 2.14. Every closed set of a 퐶(휏)-cosmic space is the intersection of 휏 many open sets. Proof: It is similar to the proof of Proposition 2.12. □ Proposition 2.15. Any two distinct points 푥 and 푦 of a 퐶(휏)- cosmic T0-space can be separated by closed sets; that is, there exist two closed sets 퐹 and 퐾 such that 푥 ∈ 퐹 , 푦 ∈ 퐾, and 퐹 ∩ 퐾 = ∅.

Proof: Let 푥 and 푦 be two distinct points of a 퐶(휏)-cosmic T0- space 푋. We prove that there exist two closed sets 퐹 and 퐾 such that 푥 ∈ 퐹 , 푦 ∈ 퐾, and 퐹 ∩ 퐾 = ∅. Since 푋 is a T0-space, there exists an 푈 such that either 푥 ∈ 푈 and 푦∈ / 푈 or 푥∈ / 푈 and 푦 ∈ 푈. Without loss of generality, we can suppose that 푥∈ / 푈 and 푦 ∈ 푈. Then Cl푋 ({푥}) ∩ 푈 = ∅. Since 푋 is 퐶(휏)-cosmic, there exists a closed set 퐾 such that 푦 ∈ 퐾 ⊆ 푈. Setting 퐹 = Cl푋 ({푥}), we have that 푥 ∈ 퐹 , 푦 ∈ 퐾, and 퐹 ∩ 퐾 = ∅. □ Definition 2.16. The 퐿푖푛푑푒푙표푓¨ 푛푢푚푏푒푟 퐿(푋) of a space 푋 (see [3]) is the least infinite cardinal number 휏 such that every open cover of 푋 has an open refinement of cardinality ≤ 휏. Proposition 2.17. For every 퐶(휏)-cosmic space 푋, we have that 퐿(푋) ≤ 휏. Proof: There exists a collection 풦 of subsets of 푋 with the prop- erties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏.

Let {푈휆 : 휆 ∈ Λ} be an open cover of 푋. We prove that 퐿(푋) ≤ 휏. For this,∪ it suffices to∪ prove that there exists a subset Λ0 of Λ such that 푈 = 푈 and ∣Λ ∣ ≤ 휏. We consider the 휆∈Λ0 휆 휆∈Λ 휆 0 family

풦0 = {퐾 ∈ 풦 : there exists 휆 ∈ Λ such that 퐾 ⊆ 푈휆}. 156 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

For every 퐾 ∈ 풦0, we choose an element 휆(퐾) of Λ with 퐾 ⊆ 푈휆(퐾). Consider the function 푓 : 풦0 → Λ given by the form 푓(퐾) = 휆(퐾). Setting Λ0 ≡ 푓(풦0) ⊆ Λ, we have

∣Λ0∣ = ∣푓(풦0)∣ ≤ ∣풦0∣ ≤ ∣풦∣ ≤ 휏. ∪ Also, if 푥 ∈ 푈 , then there exists 휆 ∈ Λ such that 푥 ∈ 푈 . 휆∈Λ 휆 0 휆0 By property (1), there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈 . 0 ∪0 휆0 Therefore, 퐾0 ∈ 풦0 and, hence, 푥 ∈ 퐾0 ⊆ 푈 ⊆ 푈휆. ∪ ∪ ∪ 휆(퐾0) ∪휆∈Λ0 Thus, 푈 ⊆ 푈 and, therefore, 푈 = 푈 . 휆∈Λ 휆 휆∈Λ0 휆 휆∈Λ0 휆 휆∈Λ 휆 □ Definition 2.18. The 푑푒푛푠푖푡푦 푑(푋) of a space 푋 (see [3]) is the least cardinal number 휏 such that 푋 has a dense subset of cardi- nality 휏. Proposition 2.19. For every 퐶(휏)-cosmic space 푋, we have that 푑(푋) ≤ 휏. Proof: There exists a collection 풦 of subsets of 푋 with the prop- erties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏.

For every 퐾 ∈ 풦, we choose a point 훼퐾 ∈ 퐾. Setting

퐷 ≡ {훼퐾 : 퐾 ∈ 풦}, we have that 퐷 is dense in 푋 and since ∣풦∣ ≤ 휏, 푑(푋) ≤ 휏. □

Notation 2.20. Let 푋 be a . By ℱ(푋), we denote the set of nonempty finite subsets of 푋.

For every finite set {퐴1, . . . , 퐴푛} of subsets of 푋, we denote by ⟨퐴1, . . . , 퐴푛⟩ the family 푛 {퐹 ∈ ℱ(푋): 퐹 ⊂ ∪푖=1퐴푖 and 퐹 ∩ 퐴푖 ∕= ∅ for 푖 = 1, . . . , 푛}.

Definition 2.21. The Vietoris topology 휏푉 on ℱ(푋) is the topology for which the family

{⟨푈1, . . . , 푈푛⟩ : 푈푖 is open in X for i = 1,..., n} is a base. Proposition 2.22. If 푋 is a 퐶(휏)-cosmic space, then the space (ℱ(푋), 휏푉 ) is 퐶(휏)-cosmic. 퐶(휏)-COSMIC SPACES 157

Proof: There exists a collection 풦 of subsets of 푋 with the prop- erties (1) for every open subset 푈 of 푋 and every 푥 ∈ 푈, there exists 퐾 ∈ 풦 such that 푥 ∈ 퐾 ⊆ 푈, and (2) ∣풦∣ ≤ 휏. We consider the family

풦0 = {⟨퐾1, 퐾2, . . . , 퐾푚⟩ : 퐾푖 ∈ 풦 for i = 1, 2,..., m}.

Obviously, the family 풦0 satisfies property (2) of Definition 2.1. We prove property (1) of Definition 2.1. Let 푊 be an open subset of ℱ(푋) and 퐹 ∈ 푊 . Clearly, 퐹 is a finite subset of 푋. Without loss of generality, we can suppose that

푊 ≡ ⟨푈1, . . . , 푈푛⟩, 푛 where 푈푖 is open in 푋 for every 푖 = 1, . . . , 푛. Then 퐹 ⊂ ∪푖=1푈푖 and 퐹 ∩ 푈푖 ∕= ∅, 푖 = 1, . . . , 푛. Let

퐹 ∩ 푈푖 = {푥푖,1, . . . , 푥푖,푚푖 }, 푖 = 1, . . . , 푛.

For every 푖 = 1, . . . , 푛, there exists 퐾푖,푗 ∈ 풦 such that

푥푖,푗 ∈ 퐾푖,푗 ⊆ 푈푖, 푗 = 1, . . . , 푚푖. We consider the element

퐾 = ⟨퐾1,1, . . . , 퐾1,푚1 , 퐾2,1, . . . , 퐾2,푚2 , . . . , 퐾푛,1, . . . , 퐾푛,푚푛 ⟩ of the family 풦0. By the above, we have 퐹 ∈ 퐾 ⊆ 푊 . Thus, the space (ℱ(푋), 휏푉 ) is 퐶(휏)-cosmic. □ Remark 2.23. We observe that Proposition 2.22 is also true if we replace the notion of 퐶(휏)-cosmic space by the notion of cosmic space. Notation 2.24. Let 푋 be a topological space and ∼ be an equiv- alence relation on 푋. We denote by C(∼) the set of all equivalence classes of 푋, by [푥] the equivalence class of 푋 with 푥 ∈ 푋, and by 푞 : 푋 → C(∼) the map for which 푞(푥) = [푥] for every 푥 ∈ 푋. The topology 휏 = {푈 ⊆ C(∼): 푞−1(푈) is open in 푋} on C(∼) is called the quotient topology and the set C(∼) equipped with it is called the quotient space. We say that an equivalence relation ∼ on a space 푋 is closed if the map 푞 is closed. We note that the equivalence relation ∼ on 158 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

푋 is closed if and only if for every closed subset 퐹 of 푋 the set ∪{[푥]: 푥 ∈ 퐹 } is closed in 푋. By Proposition 2.7 the following proposition is true. Proposition 2.25. Let 푋 be a 퐶(휏)-cosmic space and ∼ be a closed equivalence relation on 푋. Then the space C(∼) is 퐶(휏)-cosmic.

3. 퐶(휏)-cosmic spaces and universality In this section we use notions and notation from [4]. For this reason, we begin with some of them.

In what follows, all spaces are considered to be T0-spaces of weight ≤ 휏, where 휏 is a fixed infinite cardinal. We shall use the symbol “≡” in order to introduce new notations without mention of this fact. If “∼” is an equivalence relation on a non-empty set 푋, then the set of all equivalence classes of ∼ is denoted by C(∼). Let S be an indexed collection of spaces. An indexed collection 푋 M ≡ {{푈훿 : 훿 ∈ 휏} : 푋 ∈ S}, (1) 푋 where {푈훿 : 훿 ∈ 휏} is an indexed base for 푋, is called a co-mark of S. The co-mark M of S is said to be a co-extension of a co-mark + 푋 M ≡ {{푉훿 : 훿 ∈ 휏} : 푋 ∈ S} of S if there exists a one-to-one mapping 휃 of 휏 into itself such 푋 푋 that for every 푋 ∈ S and for every 훿 ∈ 휏, 푉훿 = 푈휃(훿). The corresponding mapping 휃 is called an indicial mapping from M+ to M. Let 푠 R1 ≡ {∼1: 푠 ∈ ℱ} and 푠 R0 ≡ {∼0: 푠 ∈ ℱ} be two indexed families of equivalence relations on S. It is said that R1 is a final refinement of R0 if, for every 푠 ∈ ℱ, there exists 푡 ∈ ℱ 푡 푠 such that ∼1⊆∼0. An indexed family R ≡ {∼푠: 푠 ∈ ℱ} of equivalence relations on S is said to be admissible if the following conditions are satisfied: (a) ∼∅= S × S, (b) for every 푠 ∈ ℱ, the number of ∼푠-equivalence classes is finite, and (c) ∼푠 ⊆ ∼푡, if 푡 ⊆ 푠. We denote by C(R) the 퐶(휏)-COSMIC SPACES 159 set ∪{C(∼s): 푠 ∈ ℱ}. The minimal ring of subsets of S containing C(R) is denoted by C♦(R). Consider the co-mark (1) of S. We denote by 푠 RM ≡ {∼M: 푠 ∈ ℱ} 푠 the indexed family of equivalence relations ∼M on S defined as 푠 follows: For every 푋, 푌 ∈ S, we set 푋 ∼M 푌 if and only if there exists an isomorphism 푖 of the algebra of subsets of 푋 generated by 푋 the set {푈훿 : 훿 ∈ 푠} onto the algebra of subsets of 푌 generated 푌 푋 푌 by the set {푈훿 : 훿 ∈ 푠} such that 푖(푈훿 ) = 푈훿 for every 훿 ∈ 푠. ∅ Also, we set ∼M= S × S. An admissible family R of equivalence relations on S is said to be M-admissible if 푅 is a final refinement of RM. Let R ≡ {∼푠: 푠 ∈ ℱ} be an M-admissible family of equivalence relations on S. On the set of all pairs (푥, 푋), where 푋 ∈ S and M 푥 ∈ 푋, we consider an equivalence relation, denoted by ∼R , as M 푠 follows: (푥, 푋) ∼R (푦, 푌 ) if and only if 푋 ∼ 푌 for every 푠 ∈ ℱ, 푋 푌 푋 푌 and either 푥 ∈ 푈훿 and 푦 ∈ 푈훿 or 푥∈ / 푈훿 and 푦∈ / 푈훿 for every M 훿 ∈ 휏. The set of all equivalence classes of the relation ∼R is denoted by T(M, R) or simply by T. For every H ∈ C♦(R), the set of all a ∈ T(M, R) for which there exists an element (푥, 푋) ∈ a such that 푋 ∈ H is denoted by T(H). ♦ T For every 훿 ∈ 휏 and H ∈ C (R), we denote by 푈훿 (H) the set of all a ∈ T(M, R) for which there exists an element (푥, 푋) ∈ a such 푋 that 푋 ∈ H and 푥 ∈ 푈훿 . For every subset 휅 of 휏 and L ∈ C♦(R), we set T T ♦ (1) B♦ ≡ {푈훿 (H): 훿 ∈ 휏 and H ∈ C (R)}, T T ♦ (2) B♦,휅 ≡ {푈훿 (H): 훿 ∈ 휅 and H ∈ C (R)}, L T 푇 (3) B♦,휅 ≡ {푈훿 (H) ∈ 퐵♦,휅 : H ⊆ L}. T The set B♦ is a base for a topology on the set T(M, R) such that the corresponding space is a T0-space of weight ≤ 휏. Moreover, if, 푋 for every 푋 ∈ S, the set {푈훿 : 훿 ∈ 휅} is a base for 푋, then the set T B♦,휅 is a base for the same topology on T(M, R). Therefore, the L family B♦,휅 is a base for T(L). 푋 For every element 푋 of S, there exists a natural embedding 푖T of 푋 into the space T(M, R) defined as follows: For every 푥 ∈ 푋, 푋 푖T (푥) = a, where a is the element of T(M, R) containing the pair 160 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

(푥, 푋). Thus, we have constructed a containing space T(M, R) for S of weight ≤ 휏. A class IP of spaces is said to be saturated if, for every indexed collection S of spaces belonging to IP, there exists a co-mark M+ of S satisfying the following condition: For every co-extension M of M+, there exists an M-admissible family R+ of equivalence re- lations on S such that for every admissible family R of equivalence relations on S, which is a final refinement of R+, and for every L ∈ C♦(R), the space T(L) belongs to IP. The co-mark M+ is said to be an initial co-mark of S corresponding to the class IP and the family R is said to be an initial family of S corresponding to the co-mark M and the class IP. We recall that

(a) the class of all T0-spaces of weight ≤ 휏, (b) the class of all T0 countable-dimensional spaces of weight ≤ 휏, (c) the class of all T0 strongly countable-dimensional spaces of weight ≤ 휏, (d) the class of all T0 locally finite-dimensional spaces of weight ≤ 휏, and (e) the class of all T0-spaces 푋 of weight ≤ 휏 such that ind(푋) ≤ 훼 ∈ 휏 + are saturated. It is known that if IP is a saturated class of spaces, then in IP there exists a universal element (see [4, Proposition 2.1.4]).

Proposition 3.1. The class IP of all 퐶(휏)-cosmic T0-spaces of weight ≤ 휏, where 휏 is a fixed infinite cardinal, is saturated. Proof: Let S be an indexed collection of elements of IP and

+ 푋 M ≡ {{푉훿 : 훿 ∈ 휏} : 푋 ∈ S} a co-mark of S. For every 푋 ∈ S, we denote by

푋 푋 풦 ≡ {퐾휀 : 휀 ∈ 휏} a collection of closed subsets of 푋 satisfying properties (1) and (2) of Definition 2.1. Consider an arbitrary co-mark

푋 M ≡ {{푈훿 : 훿 ∈ 휏} : 푋 ∈ S} 퐶(휏)-COSMIC SPACES 161 of S, which is a co-extension of M+. Denote by R+ any M- admissible family of equivalence relations on S. We prove that M+ is an initial co-mark of S corresponding to the class IP, and the family R+ is an initial family of equivalence relations on S cor- responding to the co-mark M and the class IP. Let R ≡ {∼푠: 푠 ∈ ℱ} be an arbitrary admissible family of equiv- alence relations on S, which is a final refinement of R+. We need to prove that T ≡ T(M, R) is a 퐶(휏)-cosmic space. For this, we consider a point a of T and an open neighborhood 푈 of a. Without T T loss of generality, we can suppose that 푈 ≡ 푈훿 (H) ∈ B♦,훿 for some 푡 푋 훿 ∈ 휏 and some H ∈ 퐶(∼ ), 푡 ∈ ℱ. Let (푥, 푋) ∈ a. Then 푥 ∈ 푈훿 푋 푋 and 푋 ∈ H. There exists 휀(훿) ∈ 휏 such that 푥 ∈ 퐾휀(훿) ⊆ 푈훿 . We consider the collection 풦푇 of all sets of the form 푇 푌 퐾휂 (E) ≡ {b ∈ T : there exists (푦, 푌 ) ∈ b with 푦 ∈ 퐾휀(휂), 푌 ∈ E}, where 휂 ∈ 휏 and E ∈ C♦(R). Then, we have 푇 T (1) a ∈ 퐾훿 (H) ⊆ 푈훿 (H), (2) ∣풦푇 ∣ ≤ 휏. 푇 By the above, it suffices to prove that the set 퐾훿 (H) is closed in 푇 T. Indeed, let b ∈ T ∖ 퐾훿 (H). We need to prove that there exists T T T 푇 푈휂 (L) ∈ B♦,훿 such that b ∈ 푈휂 (L) ⊆ T ∖ 퐾훿 (H). 푇 푌 Since b ∈/ 퐾훿 (H), for every (푦, 푌 ) ∈ b, we have 푦∈ / 퐾휀(훿) or 푌 푌∈ / H. Let (푦, 푌 ) ∈ b. There exists 휂 ∈ 휏 such that 푦 ∈ 푈휂 . For 푌 the two cases 푌∈ / H or 푦∈ / 퐾휀(훿), we have the following: If 푌∈ / H, then there exists L ∈ 퐶(∼푡) such that 푌 ∈ L and, T therefore, b ∈ 푈휂 (L). We note that H ∩ L = ∅. We prove that T 푇 T 푈휂 (L) ⊆ T ∖ 퐾훿 (H). Let c ∈ 푈휂 (L). For every (푧, 푍) ∈ c, we have 푍 ∈ L and, hence, 푍∈ / H. Thus, by the definition of the set 푇 푇 퐾훿 (H), c ∈ T ∖ 퐾훿 (H). 푌 푌 Now, we suppose that 푦∈ / 퐾휀(훿) and 푌 ∈ H. Then 푦 ∈ 푌 ∖퐾휀(훿). 푌 푌 Since the subset 푌 ∖ 퐾휀(훿) of 푌 is open and 푦 ∈ 푈휂 , there exists 휂′ ∈ 휏 such that 푌 푌 푌 푦 ∈ 푈휂′ ⊆ 푈휂 ∩ (푌 ∖ 퐾휀(훿)). 푌 T By the fact that 푦 ∈ 푈휂′ and 푌 ∈ H, we have b ∈ 푈휂′ (H). Also, 푌 푇 since 푦 ∈ 푌 ∖ 퐾휀(훿), by the definition of the set 퐾훿 (H), we have 162 D. N. GEORGIOU, S. D. ILIADIS, AND A. C. MEGARITIS

T 푇 푈휂′ (H) ⊆ T ∖ 퐾훿 (H). Thus, T 푇 b ∈ 푈휂′ (H) ⊆ T ∖ 퐾훿 (H) 푇 and, therefore, the set 퐾훿 (H) is closed in T. □

Corollary 3.2. In the class of all 퐶(휏)-cosmic T0-spaces of weight ≤ 휏, where 휏 is a fixed infinite cardinal, there exists a universal element. Using the fact that the intersection of saturated classes of spaces is saturated (see [4, Proposition 2.1.3]), we have the following corol- lary.

Corollary 3.3. Let IP휏 be the class of all 퐶(휏)-cosmic T0-spaces of weight ≤ 휏, where 휏 is a fixed infinite cardinal, and let IP be an arbitrary saturated class of spaces. Then, in the class IP휏 ∩IP, there exist universal elements. Remark 3.4. Let 푆 = {0, 1} with the topology {∅, {0}, {0, 1}} (Sierpi´nskispace). It is known that the Alexandroff cube 푆휏 is a universal space for all T0-spaces of weight ≤ 휏, where 휏 ≥ 휔. However, the Alexandroff cube 푆휏 is not 퐶(휏)-cosmic.

Let Λ be a set with∏ ∣Λ∣ = 휏, 푋휆 = 푆 for every 휆 ∈ Λ, and 휆0 ∈ Λ. Obviously, 푆휏 = 푋 . We consider the point 푥 = {푥 } of 휆∈Λ 휆 ∏ 휆 휆∈Λ 푆휏 , where 푥 = 0 for every 휆 ∈ Λ. Then 푥 ∈ {0} × 푋 , 휆 휆∈Λ∖{휆0} 휆 but there is not a closed subset 퐹 of 푆휏 such that ∏ 푥 ∈ 퐹 ⊆ {0} × 푋휆.

휆∈Λ∖{휆0} Indeed, let 퐹 be a closed subset of 푆휏 such that ∏ 푥 ∈ 퐹 ⊆ {0} × 푋휆.

휆∈Λ∖{휆0}

Then ∏ ( ∏ ) ∏ 푋휆 ∖ {0} × 푋휆 ⊆ 푋휆 ∖ 퐹

휆∈Λ 휆∈Λ∖{휆0} 휆∈Λ and ∏ ( ∏ ) {푦휆}휆∈Λ ∈ 푋휆 ∖ {0} × 푋휆 ,

휆∈Λ 휆∈Λ∖{휆0} 퐶(휏)-COSMIC SPACES 163 ∏ where 푦 = 1 for every 휆 ∈ Λ. Therefore, {푦 } ∈ 푋 ∖ 퐹 . 휆 ∏ ∏ 휆 휆∈Λ 휆∈Λ 휆 Since the subset 휆∈Λ 푋휆 ∖ 퐹 of 휆∈Λ 푋휆 is open, we have ∏ ∏ 푋휆 ∖ 퐹 = 푋휆 휆∈Λ 휆∈Λ and, hence, 퐹 = ∅, which is a contradiction. Therefore, the Alexandroff cube 푆휏 is containing space for all 퐶(휏)-cosmic spaces of weight ≤ 휏. Remark 3.5. In the class of all cosmic spaces, there does not exist a universal element. This is a well-known fact, apparently due to V. V. Uspenski˘ı(see [6]). There are 2c non-homeomorphic regular on a countable set – for example, of the form 휔 ∪ {푓}, where 푓 ∈ 훽휔 ∖ 휔. On the other hand, every cosmic space has at most c countable subspaces. By [4], this implies that the class of all cosmic spaces is not saturated. A similar argument (for spaces of cardinality 휏 rather than count- able), shows that there is no universal element in the class of 퐶(휏)- cosmic spaces for any infinite 휏.

4. Questions

1. Does there exist a T1-space 푋 of weight 휏, where 휏 is an infinite cardinal, which is not 퐶(휏)-cosmic?

2. Does there exist a T2-space 푋 of weight 휏, where 휏 is an infinite cardinal, which is not 퐶(휏)-cosmic? 3. Is Proposition ?? true for the compact open topology on the space 퐶(푌, 푍)?

Acknowledgment. We are grateful to the referee for a number of helpful suggestions for improvement of the article, particularly for Remark 2.2(11) and (12), Remark 2.11, and Remark 3.5.

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[3] Ryszard Engelking, General Topology. Translated from the Polish by the author. 2nd ed. Sigma Series in Pure Mathematics, 6. Berlin: Heldermann Verlag, 1989. [4] S. D. Iliadis, Universal Spaces and Mappings. North-Holland Mathematics Studies, 198. Amsterdam: Elsevier Science B.V., 2005.

[5] E. Michael, ℵ0-spaces, J. Math. Mech. 15 (1966), 983–1002. [6] V. V. Uspenski˘ı, Imbeddings into function spaces (Russian), Dokl. Akad. Nauk SSSR 242 (1978), no. 3, 545–548.

(Georgiou) Department of Mathematics; University of Patras; 26504 Patras, Greece E-mail address: [email protected]

(Iliadis) Department of Mathematics; University of Patras; 26504 Patras, Greece E-mail address: [email protected]

(Megaritis) Department of Mathematics; University of Patras; 26504 Patras, Greece E-mail address: [email protected]