Spaces, Topological Games, and Selection Principles

Volume 36, 2010 Pages 107–122

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

퐷-Spaces, Topological Games, and Selection Principles

by Leandro F. Aurichi

Electronically published on February 5, 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 36 (2010) Pages 107-122 E-Published on February 5, 2010

퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES

LEANDRO F. AURICHI

Abstract. We present some connections between selection principles and the 퐷-space problem. Our main result is that every Menger space is a 퐷-space. We also present a study of the Rothberger property restricted to compact spaces.

1. Introduction The definition of a 퐷-space appeared in [7]. There, it was also asked if it is true that every Lindelöfspace is a 퐷-space. Many classes of spaces were proven to be included in the class of 퐷-spaces (see, e.g., [1], [6], [5], [9]), but the question about Lindelöfspaces is still open. In [8], it was said that one of the main problems about this question is the lack of covering properties that imply being a 퐷-space. In this work, we try to make a step in this direction, proving that a selection principle for open covers implies the 퐷 property. Such selection principles have been extensively studied, see, e.g., [16], [4], thus these results can give some new light to this

2010 Mathematics Subject Classification. Primary 54D99; Secondary 54A35, 54G12. Key words and phrases. 퐷-spaces; discrete subsets; Menger; Rothberger. This work was supported by CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico)and CAPES (Coordena¸cãode Aperfei¸coamento de Pessoal de N´ıvel Superior), Brazil. This paper is part of the author’s doctoral thesis under supervision of Prof. Lucia Junqueira and was done during a visit to the Mathematics Department of the University of Toronto. During this time the author was under supervision of Prof. Franklin D. Tall. The author thanks him for all his assistance during the preparation of this paper. ⃝c 2010 Topology Proceedings. 107 108 L. F. AURICHI problem. Following this idea, we present some topological games that are related to selection principles and give the relation between them and 퐷-spaces. In [3], it was shown that, under MA, Lindelöfspaces that are a union of fewer than continuum compact spaces are 퐷-spaces. This can be obtained by our main result simply by using the fact that such spaces have the Menger property. All the definitions about selection principles can be found in [16]. We finish this section defining 퐷-space, fixing some notation, and proving a simple result.

Deﬁnition 1.1. Let 푋 be a topological space. We say that (푉푥)푥∈푋 is an open neighborhood assignment for 푋 if each 푉푥 is∪ an open neighborhood of 푥. If 푌 ⊂ 푋, we denote by 푉푌 the set 푦∈푌 푉푦. We say that 푌 is a kernel for (푉푥)푥∈푋 if 푉푌 = 푋. Finally, we say that 푋 is a 퐷-space if, for every open neighborhood assignment (푉푥)푥∈푋 , there is a closed discrete kernel. In [5], it was proven that every strong Σ-space is a 퐷-space. In particular, this result implies that every Moore space is a 퐷-space. A Moore space is a regular developable space. We will present here a direct proof which also shows that the regularity in this case is not needed. Before this, we give some deﬁnitions.

Deﬁnition 1.2. Let 푋 be a topological space, 풞 a∪ family of open sets, and 푥 ∈ 푋. We use 푠푡(푥, 풞) to denote the set {퐶 ∈ 풞 : 푥 ∈ 퐶}. We say that 푋 is a developable space if there is a sequence of open covers (풞푛)푛∈휔 such that for each 푥 ∈ 푋, {푠푡(푥, 풞푛): 푛 ∈ 휔} is a local base for it. Proposition 1.3. Every developable space is a 퐷-space.

Proof: Let 푋 be a developable space and let (풞푛)푛∈휔 be a sequence of covers such that for each 푥 ∈ 푋, {푠푡(푥, 풞푛): 푛 ∈ 휔} is a local base for it. Let (푉푥)푥∈푋 be an open neighborhood assignment for 푋. For each 푛 ∈ 휔, let 푋푛 = {푥∪∈ 푋 : 푠푡(푥, 풞푛) ⊂ 푉푥}. This set can be empty, but note that 푋 = 푛∈휔 푋푛. For each 푛 ∈ 휔, we will define 퐷푛 ⊂ 푋푛 satisfying the following: (a) 퐷푛 is closed∪ discrete; (b) 푉 ∪ 푉 ⊃ 푋 . 퐷푛 푘<푛 퐷푘 푛 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 109 ∪ Well order 푋 ∖ 푉 and define 푑 as the least element of ∪ 푛 푘<푛 퐷푘 휉 (푋 ∖ 푉 ) ∖ 푉 where 퐸 = {푑 : 휂 < 휉}. Define 퐷 as 푛 푘<푛 퐷푘 퐸휉 휉 휂 푛 the set of such 푑휉’s. We will prove that 퐷푛 is closed discrete. Let 푦 ∈ 푋. Let 퐶 ∈ 풞푛 such that 푦 ∈ 퐶. Suppose there is a 푑휉 ∈ 퐶. Pick the least one. Note that 푦 ∈ 푠푡(푑 , 퐶 ). Thus, 푦 ∈ 푉 ∩ 퐶 휉 푛 푑휉 and 푉 ∩ 퐶 ∩ 퐷 = {푑 }. 푑휉 푛 ∪ 휉 Therefore, 퐷 = 푛∈휔 퐷푛 is such that 푉퐷 = 푋 and it is closed discrete. □

2. The partial open neighborhood assignment game Through this work we will deal with some games that are played by two players. In general, one of the players is looking for a cover of a space and the other one is not. For mnemonic reasons, we will name these players 퐶 and 푁, respectively. The general idea for the first game is the following: player 푁 gives a partial open neighborhood assignment for the space that is enough to cover it. Then player 퐶 chooses a subset of the domain of this partial open neighborhood assignment that will be a part of the closed discrete kernel that 퐶 is looking for. Then player 푁 chooses another partial open neighborhood assignment and player 퐶 chooses another part for its closed discrete kernel. Definition 2.1. Let 푋 be a topological space. We call the following game between players 푁 and 퐶 the partial open neighborhood assignment game (PONAG). Player 푁 defines a partial open neighborhood assignment (푉푥)푥∈푌0 for 푋 with 푌0 ⊂ 푋 and such that

푉푌0 = 푋. After this, player 퐶 chooses 퐷0 ⊂ 푌0, a closed discrete subset of 푋. Then, at the 푛-th inning,

∙ 푁 chooses (푉푥)푥∈푌푛 , a partial open neighborhood assignment compatible with every (푉 ) for 푘 < 푛 such that ∪ 푥 푥∈푌푘 – it covers 푋 ∖ 푉 and ∪ 푘<푛 퐷푘 – 푌 ∩ 푉 = ∅; 푛 푘<푛 퐷푘 ∙ 퐶 chooses 퐷 ⊂ 푌 , a closed discrete subset of 푋. 푛 푛 ∪ We say that 퐶 wins the game if 푛∈휔 푉퐷푛 = 푋. One of the advantages of this game is that we do not have to care about making the kernel closed discrete.

Proposition∪ 2.2. Let (퐷푛)푛∈휔 be as in a PONAG won by player 퐶. Then 푛∈휔 퐷푛 is a closed discrete set. 110 L. F. AURICHI

∪ Proof: Let 푥 ∈ 푋,∪ and let 푛 ∈ 휔 be the ﬁrst such that 푥 ∈ 푉 . Note that 푉 is an open set that separates 푥 푑∈퐷푛 푑 ∪ 푑∈퐷푛 푑 ∪ from every point in 푘>푛 퐷푘. Since 푘≤푛 퐷푘 is closed discrete (it is a ﬁnite∪ union of closed discrete sets), 푥 is not an accumulation point of 푘∈휔 퐷푘. □ It is easy to see that being a 퐷-space is related to player 푁 not having a winning strategy. Proposition 2.3. Let 푋 be a topological space. If 푁 has no winning strategy for the PONAG, then 푋 is a 퐷-space.

Proof: Suppose 푋 is not a 퐷-space. Let (푉푥)푥∈푋 be an open neighborhood assignment for 푋 that witnesses∪ it. Then, if 푁 plays at every inning 푛,(푉 ) where 퐴 = 푋 ∖ 푉 , 퐶 cannot 푥 푥∈퐴 푘<푛 퐷푘 win this game. □ It is easy to prove that for a 휎-compact space, player 퐶 has a winning strategy for the PONAG. There is a generalization for 휎- compactness that we can prove which implies that player 푁 has no winning strategy for the PONAG. Before proving this, we give some definitions. Definition 2.4. We say that a space 푋 is a Menger space if for every sequence (풰푛)푛∈휔 of open covers∪ there∪ is (푈푛)푛∈휔 such that each 푈푛 is a finite subset of 풰푛 and 푛∈휔 푈푛 = 푋. Note that the Menger property implies the Lindelöfproperty; thus, we can assume that each cover in the definition of the Menger property is countable. Also, it is easy to see that the previous definition has an equivalent formulation if we also require each 풰푛 to be closed under finite unions and each 푈푛 to be a unitary subset of 풰푛. The property of being a Menger space has an equivalent formulation using a game. Definition 2.5. Let 푋 be a Lindelöfspace. We call the Menger game the following game played between players 푁 and 퐶. At each inning 푛 ∈ 휔, player 푁 chooses a countable open cover 풰푛 that is closed under finite unions. Then player∪ 퐶 chooses 푈푛 ∈ 풰푛. We say that player 퐶 wins the game if 푛∈휔 푈푛 = 푋. 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 111

In [11], it was proven that a Lindelöfspace is a Menger space if and only if player 푁 has no winning strategy for the Menger game. With this result, we can prove the following proposition. Proposition 2.6. Let 푋 be a Menger space. Then 푁 has no winning strategy for the PONAG. Proof: Suppose that 푁 has a winning strategy for the PONAG. We will define a winning strategy for 푁 for the Menger game. Since 푋 is Menger, this will give us a contradiction. At the first inning,

푁 chooses 풰0 = {푉푥 : 푥 ∈ 푌0} where (푉푥)푥∈푌0 is the one chosen by 푁 at the ﬁrst play in the winning strategy for the PONAG. If

(푉푥)푥∈푈0 is the ﬁnite subset of (푉푥)푥∈푌0 played by 퐶 in the Menger game, then consider 퐶 in PONAG as playing the ﬁnite set 푈0 ⊂ 푌0.

Then at the second inning, 푁 chooses 풰1 = {푉푈0 ∪ 푉푥 : 푥 ∈ 푌1} where 푌1 is the one that 푁 chooses in the winning strategy for the

PONAG after 퐶 chooses (푉푥)푥∈푈0 . Note that proceeding this way, 푁 wins the game. □ Corollary 2.7. Every Menger space is a 퐷-space. But the PONAG does not give us an equivalence for being a 퐷-space. The space of the irrationals with the usual topology is a 퐷-space (see, e.g., [1]), and we can prove the following proposition. Proposition 2.8. In the PONAG for 휔휔, player 푁 has a winning strategy.

푛 휔 Proof: For each 푛 ∈ 휔, let 퐴 = {푦 ∈ 휔 : 푗 ∈ 휔} and (푉 푛 ) 푛 푗 푦푗 푗∈휔 be such that { 푛 + 1 if 푚 ≤ 푗 ∙ 푦푛(푚) = 푗 0 otherwise 휔 ∙ 푉 푛 = {푥 ∈ 휔 : 푥(푛) ≤ 푛 + 1 + 푗}. 푦푗 푛 Note that 퐴 ∩ 퐴 = ∅ if 푛 ∕= 푘 and 푦 ∈ 푉 푛 for any 푗, 푛 ∈ 휔. 푛 푘 푗 푦푗 푛 Note also that (푦푗 )푗∈휔 is a converging sequence to the function constantly equal to 푛 + 1.

At the first inning, 푁 chooses 푉푦∈푌0 where 푌0 = 퐴0. Note that player 퐶 can choose only finitely many points of 푌0. Let 푈0 be the 0 0 푦푗 ’s chosen. Let 푘 = max{푗 + 1 : 푦푗 ∈ 푈0}. Then player 푁 chooses 휔 푌1 = 퐴푘+1. Note that 푉푌1 = 휔 and 푌1 ∩ 푉푈0 = ∅. Again, player 퐶 can choose only finitely many points in 푌1. Let 푈1 be the set 112 L. F. AURICHI

푘+1 ′ 푘+1 of the 푦푗 ’s chosen. Let 푘 = max{푗 + 푘 + 1 : 푦푗 ∈ 푈1}. Then player 푁 chooses 푌2 = 퐴푘′+1. Proceeding this way, player 푁 wins the game. □

3. The star game The next game is based on the proof in [1] that every space with a point countable base is a 퐷-space. Deﬁnition 3.1. Let 푋 be a topological space. We call the star game the following game played between 퐶 and 푁. At each inning 휉, player 퐶 chooses 푥휉 diﬀerent from every 푥훼 chosen before. Then player 푁 chooses 푆휉 ⊂ 푋 such that 푥휉 ∈ 푆휉 and for every 푦 ∈ 푆휉, 푁 chooses an open set 푉푦 such that 푦, 푥휉 ∈ 푉푦. Also, if 푦 ∈ 푆휂 for some 휂 < 휉, then 푉푦 has to be the same one previously chosen. At the beginning of every inning, the following tests are made in this order: (a) If there is an 푥 ∈ 푋 such that 푥 is an accumulation point for {푥휂 : 휂 < 휉} and 푥 is not an element of any 푆휂 with 휂 < 휉, then 푁 loses. We call this condition 푆. (b) If {∪푥휂 : 휂 < 휉} is not a closed discrete set, then 퐶 loses. (c) If 휂<휉 푉푥휂 = 푋, then 퐶 wins. Otherwise, the game continues. Proposition 3.2. Let 푋 be a topological space. If 푁 has no winning strategy for the star game, then 푋 is a 퐷-space.

Proof: Suppose that 푋 is not a 퐷-space. Let (푉푥)푥∈푋 be an open neighborhood assignment witnessing it. If 푁 plays at every inning 휉, 푆휉 = {푦 : 푥휉 ∈ 푉푦}, then player 푁 wins. □ For 휎-compact spaces, the situation is similar to the PONAG. Proposition 3.3. If 푋 is a 휎-compact space, then player 퐶 has a winning strategy for the star game. Proof: First, suppose 푋 is compact. Player 퐶 plays as follows. Define 푥0 as any∪ point of 푋. Next, define 푥푛+1 as any point (if there is any) not in 푚<푛 푉푆푚 . Note that if this procedure continues until the 휔-th inning,∪ then any accumulation point of {푥푛 : 푛 ∈ 휔} must be outside 푆 . Thus, unless player 푁 loses because of 푛∈휔 푛 ∪ condition 푆, there is an 푛 ∈ 휔 such that 푚<푛 푉푆푚 covers 푋. Since 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 113 ∪ 푋 is compact, we can find a finite subset 퐹 of 푚<푛 푆푚 ∖ {푥푚 : 푚 < 푛} such that 푉 ∪ 푉 covers 푋. 퐹 {푥푚:푚<푛} ∪ For proving the 휎-compact case, just write 푋 = 푛∈휔 퐾푛 with each 퐾푛 compact and repeat the strategy above countably many times. □ It is not difficult to see that a stronger version of the Menger property implies that player 푁 has no winning strategy for the star game. Before proving this, let us give some definitions. Definition 3.4. We say that a space 푋 is a Rothberger space if for every sequence (풰푛)푛∈휔 of open covers of 푋 there is a cover (푈푛)푛∈휔 for 푋 such that each 푈푛 ∈ 풰푛. Note that the Rothberger property implies Lindelöfness. Thus, we can assume again that all the covers are countable. Note also that the Rothberger property implies the Menger property. Sim- ilar to the Menger property, the Rothberger property also has a characterization using a game. Definition 3.5. Let 푋 be a topological space. The point-open game is the following game played between players 푁 and 퐶. At each inning 푛 ∈ 휔, player 퐶 chooses 푥푛 ∈ 푋 and player∪ 푁 chooses an open neighborhood 푉푛 for 푥푛. Player 퐶 wins if 푛∈휔 푉푛 = 푋. In [14], it was proven that being a Rothberger space is equivalent to player 푁 not having a winning strategy for the point-open game. Therefore, if 푋 is a Rothberger space, player 푁 cannot have a winning strategy for the star game, because, until the 휔-th inning, this game is played like the point-open game, just with more restrictions for player 푁. Proposition 3.6. If 푋 is a Rothberger space, then player 푁 has no winning strategy for the star game. In [1], in the proof that every space with a point countable base is a 퐷-space, actually it was proven that a more general class of spaces (a simple generalization of the concept of having an uniform base) is 퐷. Definition 3.7. We say that a base ℬ for a topological space 푋 is 휔-uniform if for every 푥 ∈ 푋 and every 퐵 ∈ ℬ such that 푥 ∈ 퐵, the set {퐴 ∈ ℬ : 푥 ∈ 퐴 and 퐴 ∕⊂ 퐵} is countable. 114 L. F. AURICHI

Note that any point countable base is an 휔-uniform base. But the converse is not true, even for Lindelöfspaces: let 퐷 be a discrete space of cardinality 휔1 and add a point ∞ such that a local base for it is made up of sets of the form 푉 ∪{∞} with 퐷 ∖푉 countable. The space 퐷 ∪ {∞} is a Lindelöfspace with an 휔-uniform base but without a point countable base. We will prove that having an 휔-uniform base is enough for a space to have a winning strategy for player 퐶 for the star game. Before this, we need some more definitions. Definition 3.8. Let 푋 be a topological space. We say that an inning 훼 for the star game is a closed inning if

∙ 퐷훼 = {푥휉 : 휉 < 훼} is a closed discrete set in 푋; ∙ 푥 ∈/ 푉 for any 휂 < 휉 < 훼; 휉 푥∪휂 ∙ 푉퐷훼 ⊃ 휉<훼 푆휉 (푆휉 is the one in Deﬁnition 3.1). We say that player 퐶 has a partial strategy∪ for the star game if, for every closed inning 훼 and every 푥∈ / 푉 , there is a closed 휉<훼∪ 푥휉 inning 훽 such that {푥 : 훼 ≤ 휉 < 훽} ⊂ 푋 ∖ 푉 and 푥 = 푥. 휉 휉<훼 푥휉 훼 The idea of the closed inning is to split a strategy into smaller parts. Proposition 3.9. Let 푋 be a topological space. If player 퐶 has a partial strategy for the star game, then player 퐶 has a winning strategy for the star game. Proof: The strategy for player 퐶 will simply be a sequence of partial strategies concatenated. At each inning 훼, player 퐶 does the following: ∙ if 퐶 is in the middle of a partial strategy, 퐶 continues∪ it; ∙ otherwise, we assume that 훼 is a closed inning. If 푉 = 휉<훼 푥휉 푋, the game is ﬁnished. Otherwise,∪ player 퐶 begins a new partial strategy for an 푥∈ / 푉 . 휉<훼 푥휉 Let us prove that this is a winning strategy. For this, it is enough to show that a limit of closed innings is also a closed inning. Let 훼 be a limit of closed innings and let 푥 ∈ {푥휉 : 휉 < 훼}. We can suppose that 푥 ∈ 푆휉 for some 휉 < 훼 (otherwise, player 푁 loses). Thus, there is a 훽 < 훼 such that 푥 ∈ 푉 (because of the partial strategies). 푥훽 We may assume by induction that {푥휉 : 휉 < 훽} is closed discrete. 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 115

Since 푥 ∈/ 푉 for 휂 > 훽, 푥 cannot be an accumulation point of 휂 푥훽 {푥휉 : 휉 < 훼}. □ Now we just have to prove that having an 휔-uniform base is enough to have a partial strategy. Proposition 3.10. If 푋 has an 휔-uniform base, then player 퐶 has a partial strategy for the star game. Proof: Let ℬ be an 휔-uniform base for 푋. Since having an 휔- uniform base is hereditary, we just have to prove that for any 푥 there is a closed inning that covers 푥. We may suppose that all 푉푥 belong to ℬ. Let 푥 ∈ 푋. We will deﬁne a strategy for player 퐶 such that 휔 will be a closed inning. For technical reasons, we will enumerate the innings for this strategy using numbers of the form 푛 푝 where 푝 is a prime number and 푛 > 0. Deﬁne 푥2 = 푥. After player 푁 plays, enumerate the set {푈 ∈ ℬ : ∃푦 ∈ 푆2 푥2 ∈ 푉푦 and 푥2 푚 푈 ∖푉푥2 ∕= ∅} as (푈푛 )푛∈휔. Assume that for each 푘 < 푛, with 푘 = 푝 for some prime 푝, player 퐶 already chose 푥푘 and enumerated the set {푈 ∈ ℬ : ∃푦 ∈ 푆 푥 ∈ 푉 and 푈 ∖ 푉 ∕= ∅} as (푈 푥푘 ) . At 0 푘 푦 푥푘 푛 푛∈휔 the 푛-th inning, with 푛 = 푝푚 for some prime 푝, player 퐶 does the 푠 following. Let 푞 be the 푚-th number∪ of the form 푟 with 푟 being a prime number. If there is an 푥 ∈ 푘<푛 푆푘 such that ∪ (3.1) 푉 ∈ {푈 푥푞 : 푡 ∈ 휔} and 푥∈ / 푉 , 푥 푡 푥푘 푘<푛

푥푞 then player 퐶 chooses 푥푛 as the 푥 such that 푉푥 = 푈푡 with 푡 being 푥푛 the least one satisfying (3.1). Enumerate (푈푡 )푡∈휔 as usual. If there is no such 푥 satisfying (3.1), proceed to the next inning without choosing an 푥 (actually, the definition of the game does not allow this, but there is no problem if we just assume that player 푁 will do nothing if player 퐶 does nothing). Let∪ us prove that∪ this is a partial strategy. We begin proving that 푛∈휔 푆푛 ⊂ 푛<휔 푉푥푛 . Let 푥 ∈ 푆푛 for some 푛 ∈ 휔. Assume 푥푛 that 푛 is the least one with this property. Then 푉푥 = 푈푘 for some 푘 ∈ 휔. Note that in our construction, at every inning of the form 푝푡, where 푝 is a prime number and 푡 indicates how many 푥푛 elements there are before 푛 in our indexes, we pick the first 푈푠 that has some element not covered. Since there are only finitely 푥푛 many elements before 푘, 푈푘 will be covered in some inning and, 116 L. F. AURICHI therefore, 푥∪. Now we have to∪ prove that {푥푛 : 푛 ∈ 휔} is closed discrete in 푛∈휔 푆푛. Let 푥 ∈ 푛∈휔 푆푛. Let 푛 be the first one such that 푥 ∈ 푉푥푛 . Since there are only finitely many 푥푘’s with 푘 < 푛 and no 푥푚 ∈ 푉푥푛 for 푚 > 푛, we are done. □ Corollary 3.11. If 푋 has an 휔-uniform base, then player 퐶 has a winning strategy for the star game. Therefore, 푋 is a 퐷-space. The relation between the star game and the 퐷 property is not clear. We do not have any example of a 퐷-space such that player 푁 has a winning strategy for the star game. Since it is not known if a hereditarily Lindelöfspace must be a 퐷-space, the following question is particularly interesting. Question 3.12. If 푋 is a hereditarily Lindelöfspace, can player 푁 have a winning strategy for the star game?

4. Some applications It is easy to see that if 푋 is a Menger space and ℙ is a countably closed forcing, then if ℙ preserves the Lindelöfnessof 푋, it has also to preserve its property of being Menger. We can prove an analogous result for 퐷-spaces such that 푁 has no winning strategy for the PONAG. Before proving this, we prove a lemma. Lemma 4.1. Let 푋 be a topological space and ℙ be a countably ˙ closed forcing. Let 푝 ∈ ℙ and (푉푥)푥∈푋 be such that 푝 ⊩“(푉푥)푥∈푋 is an open neighborhood assignment for 푋ˇ” and 푝 ⊩ 푋ˇ is Lindelöf. Let 푈 ⊂ 푋 be an open set. Then there are 푞 ≤ 푝, a countable ˙ ˇ ˇ ˇ ˙ 퐴 ⊂ 푋 ∖ 푈, and (푉푥)푥∈퐴 such that 푞 ⊩“푉퐴 ∪ 푈 = 푋 and 푉푥 = 푉푥 for every 푥 ∈ 퐴.” Proof: Note that 푝 ⊩ 푋ˇ ∖ 푈 is Lindelöf.Thus, 푝 ⊩ “there is a ˙ ˇ ˇ ˙ ˇ ′ countable 퐴 such that 푉퐴 ⊃ 푋 ∖ 푈 and 퐴 ∩ 푈 = ∅.” Let 푞 ≤ 푝 be such that 푞′ decides 퐴˙ and let 푞 ≤ 푞′ be such that 푞 decides ˙ (푉푥)푥∈퐴. □ Proposition 4.2. Let 푋 be a topological space such that 푁 has no winning strategy for the PONAG. Let ℙ be a countably closed forcing such that ℙ ⊩ 푋ˇ is Lindelöf.Then ℙ ⊩ 푋ˇ is a 퐷-space. Proof: We will define a strategy for 푁 and, for each possible move of 푁, we will associate a 푝 ∈ ℙ in the following way: 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 117

∙ In the ﬁrst inning, 푁 chooses (푉푥)푥∈퐴, where 퐴 is the one given by the lemma applied to 1 and the empty set. The 푞 given by the lemma is the element associated with this play.

∙ Let (푉푥)푥∈푌 be the move of 푁 at the 푛-th inning. Let 푝 be the element associated with this play and 퐷 ⊂ 푌 be the closed discrete set chosen by player 퐶. At the (푛 + 1)-th inning, player 푁 chooses (푉푥)푥∈퐴 where 퐴 is the one given by the lemma applied to 푝 and 푈, where 푈 is the union of every 푉퐷 already played by player 퐶. The forcing condition associated with this move is the 푞 ≤ 푝 given by the lemma.

Since 푁 has no winning strategy, there is (퐷푛∪)푛∈휔 where 퐷푛 is the one chosen by 퐶 at the 푛-th inning such that 푉 = 푋. Note ∪ 푛∈휔 퐷푛 that 퐷 = 푛∈휔 퐷푛 is closed discrete. Let (푝푛)푛∈휔 be the forcing conditions associated with every move of 푁 in this game. Note that ˇ 푝푛+1 ≤ 푝푛. Let 푝 ≤ 푝푛 for every 푛. Note that 푝 ⊩ 푉퐷 = 푋 and ˇ ˙ ˇ 푉푥 = 푉푥 for every 푥 ∈ 퐷. So 푝 forces that 푋 is a 퐷-space. □ One of the difficulties in proving that a space 푋 is a 퐷-space is determining where to look for the closed discrete kernel. For a fixed open neighborhood assignment (푉푥)푥∈푋 , one can easily fix a countable set to look at if 푋 is Lindelöf. Namely, pick 푀 a countable elementary submodel such that (푉푥)푥∈푋 ∈ 푀. It is easy to see that if (푉푥)푥∈푋 has a closed discrete kernel, then it has one as a subset of 푀 ∩ 푋. But if we assume that 푋 is a Menger space, we can do a little better.

Proposition 4.3. Let 푋 be a Menger space and (푉푥)푥∈푋 be an open neighborhood assignment for it. If 푌 ⊂ 푋 is such that for every finite 퐹 ⊂ 푌 , 푉 ∪ 푉 = 푋, then there is a closed discrete (in 퐹 푌 ∖푉퐹 푋) 퐷 ⊂ 푌 such that 푉퐷 = 푋. Proof: Suppose not. Consider the following strategy for player 푁∪for the Menger game. At∪ each inning, player 푁 defines 풰푛 = { 푚<푛 푈푚 ∪ 푉퐹 : 퐹 ⊂ 푌 ∖ 푚<푛 푈푚 is finite} where the 푈푚’s are the plays of player 퐶. Since each 푈푚 is a finite union of 푉푥’s with 푥 ∈ 푌 and by our hypothesis over 푌∪, 풰푛 is a cover for 푋∪. As in the case of the PONAG, note that if 푛∈휔 푈푛 = 푋, then 푛∈휔 퐹푛, where each 푈푛 = 푉퐹푛 , is a closed discrete set. Since there is no 118 L. F. AURICHI closed discrete set inside 푌 that covers 푋, then player 푁 wins the game, contradicting the fact that 푋 is a Menger space. □ Showing that a given space is a Menger space is often easier than proving it is a 퐷-space directly. One example is a Souslin tree with the fine wedge topology or the coarse wedge topology [2]. Such spaces are easily shown to be Rothberger by a combinatorial argument. Therefore, they are Menger spaces and 퐷-spaces. This is an easier proof than the one presented in [15], where the 퐷 property is shown directly. Also, proving that a space is a 퐷-space by proving it is a Menger space is a technique that applies to a wider class of spaces than the class of metrizable spaces or 휎-compact spaces, as we can see in the next example. Example 4.4. Let 푋 be a Menger space and 휅 be any uncountable cardinal. Define 푌 ′ as the disjoint union of 휅 many copies of 푋 and let 푌 be a one-point Lindelöficationof 푌 ′. Note that 푌 is a Menger space and it is neither metrizable nor 휎-compact.

5. Compact Rothberger spaces The Rothberger property restricted to compact spaces has sev- eral characterizations. Some of them are well known and others were presented somewhere else. For the convenience of the reader, we will present here a list of some of them, putting together new and old ones. Before this, we need the following definitions. Definition 5.1. Let 푋 be a Lindelöfspace. We call the open-open game the following game played between players 푁 and 퐶. At each inning 푛 ∈ 휔, player 푁 chooses 풰푛, a countable open cover for 푋. Then player∪ 퐶 chooses 푈푛 ∈ 풰푛. We say that player 퐶 wins the game if 푛∈휔 푈푛 = 푋. In [14], it was proven that 푋 is Rothberger if and only if player 푁 has no winning strategy for the open-open game. In [17], a characterization for preservation of the Lindelöfnessof a space under countably closed forcing was proven. This characterization uses the concept of a covering tree and, using the characterization of being a Rothberger space by topological games, it is simple to see that Rothberger spaces have the preservation property. It 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 119 turns out that the same technique gives us a characterization in the class of compact spaces and preservation under any forcing. This result also can be proven using a combination of older results, but this proof seems to have its own interest. The next definitions are versions of concepts presented in [17]. Definition 5.2. Let 푋 be a compact space. We call a covering 휔-tree a family (푉푠)푠∈휔<휔 where each 푉푠 is an open set of 푋 and <휔 for every 푠 ∈ 휔 , {푉푠⌢푛 : 푛 ∈ 휔} is a cover for 푋. The order on <휔 <휔 {푉푠 : 푠 ∈ 휔 } is the one induced by the usual one on 휔 .

Definition 5.3. Let 푋 be a compact space and (푉푠)푠∈휔<휔 be a <휔 covering 휔-tree for 푋. We∪ say that 푉푡 with 푡 ∈ 휔 is a covering element for (푉푠)푠∈휔<휔 if 푎≤푡 푉푎 ⊃ 푋. There is a characterization using elementary submodels and a special space defined using them [13]. Definition 5.4. Let (푋, 휏) be a topological space and 푀 be an elementary submodel such that (푋, 휏) ∈ 푀. We call 푋푀 the space 푋 ∩ 푀 with the topology generated by {푈 ∩ 푀 : 푈 ∈ 휏 ∩ 푀}. Proposition 5.5. Let 푋 be a compact Hausdorff space. The following are equivalent. (a) 푋 is scattered; (b) 푋 is Rothberger; (c) the compactness of 푋 is preserved by any forcing; (d) the compactness of 푋 is preserved by adjoining a Cohen real; (e) for every covering 휔-tree for 푋, the set of the covering elements is dense; (f) for every covering 휔-tree for 푋, there is a covering element; (g) there is a countable elementary submodel 푀 such that 푋푀 is compact Hausdorff; (h) 푋푀 is compact Hausdorff for any elementary submodel 푀 such that 푋 ∈ 푀. Proof: (푐) ⇒ (푑) is trivial. For (푑) ⇒ (푐), let ℙ be a forcing and let 퐶˙ be a name for an open cover for 푋 such that ℙ ⊩“퐶˙ has no finite subcover.” We may assume as well that ℙ forces that every element of 퐶˙ is an 120 L. F. AURICHI open set of the ground model. We construct a tree ((푝푠, 푉푠))푠∈휔<휔 such that, for every 푠, 푡 ∈ 휔<휔,

(i) 푝푠 ∈ ℙ and 푝푠 ≤ 푝푡 if 푠 ⊃ 푡; ˙ (ii) 푝푠 ⊩ 푉푠 ∈ 퐶; (iii) {푉푠⌢푛 : 푛 ∈ 휔} is a cover for 푋. Note that ((푝푠, 푉푠))푠∈휔<휔 is isomorphic to Cohen forcing. Let 푔 be a Cohen real in the∪ extension after we force with such order. Note that by genericity 푠⊂푔 푉푠 is an open cover for 푋. Since we have (푑), there is a ﬁnite subcover for it, although ℙ forces that there is not.

For (푑) ⇒ (푒), let (푉푠)푠∈휔<휔 be a covering 휔-tree for 푋. Suppose <휔 that there is an 푠 ∈ 휔 such that there is no 푡 ⊃ 푠 such that 푉푡 is a covering element. Let 푔 be a Cohen real extending 푠. By (푑) there is a 푠 ⊂ 푡 ⊂ 푔 such that 푉푠 is a covering element. For (푒) ⇒ (푑), suppose that Cohen forcing destroys 푋. Proceed as we did in the proof of (푑) ⇒ (푐) and construct a covering 휔-tree. Since we have (푒), any generic cover has a finite subcover. (푒) ⇒ (푓) is immediate. <휔 For (푓) ⇒ (푒), let (푉푠)푠∈휔<휔 be a covering 휔-tree and let 푡 ∈ 휔 . <휔 Note that {푉푠 : 푠 ∈ 휔 , 푠 ⊋ 푡} is isomorphic to a covering 휔-tree for 푋 and, by (푓), it has a covering element. Note that this covering element is an extension of 푉푡. Note that a strategy for player 푁 on the open-open game is a covering 휔-tree and the converse is also true: any covering 휔- tree is a strategy for the game. Also, note that such tree has a covering element if and only if the strategy associated is not a winning strategy. Thus, we have (푏) ⇔ (푓). The equivalence among (푎), (푔), and (ℎ) is done in [13]. It is well known that in the class of compact spaces the notions of being Rothberger and being scattered are equivalent. For the convenience of the reader, we present one proof of this result here. (푎) ⇒ (푏): Suppose 푋 is scattered. Let 훼 be the height of 푋 (see [10, p. 350] for the definitions of height and 푋훼). We will prove this by induction over 훼. Note that, since 푋 is compact, 훽 훼 = 훽 + 1 with 푋 finite. Let (풰푛)푛∈휔 be open covers for 푋. 훽 Let {푥0, ..., 푥푛} = 푋 . Choose∪ 푈푘 ∈ 풰푘 for 푘 = 0, ..., 푛 such that 푥푘 ∈ 푈푘. Note that 푋 ∖ 푘≤푛 푈푘 is a compact scattered space with 퐷-SPACES, TOPOLOGICAL GAMES, AND SELECTION PRINCIPLES 121 height less than 훽. Thus, by the∪ induction hypothesis,∪ there are 푈푘 ∈ 풰푘 with 푘 > 푛 such that 푘>푛 푈푘 ⊃ 푋 ∖ 푘≤푛 푈푘. Thus, 푋 is Rothberger. (푏) ⇒ (푎): We only have to prove that any closed subspace of 푋 has an isolated point. Suppose not. Since any compact space without isolated points has 2휔 as a closed subspace, and 2휔 is easily seen not to be Rothberger, we have a contradiction. □ There is a direct proof of (푎) ⇔ (푐) in [12].

References [1] A. V. Arhangel’skii and R. Z. Buzyakova, Addition theorems and 퐷-spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 4, 653–663. [2] Leandro Fiorini Aurichi, Examples from trees, related to discrete subsets, pseudo-radiality and 휔-boundedness, Topology Appl. 156 (2009), no. 4, 775–782. [3] Leandro F. Aurichi, LúciaR. Junqueira, and Paul B. Larson, 퐷-spaces, irreducibility and trees, Topology Proc. 35 (2010), 73–82. [4] L. Babinkostova and M. Scheepers, Combinatorics of open covers (IX): Basis properties, Note Mat. 22 (2003/04), no. 2, 167–178. [5] Raushan Z. Buzyakova, On 퐷-property of strong Σ spaces, Comment. Math. Univ. Carolin. 43 (2002), no. 3, 493–495. [6] , Hereditary 퐷-property of function spaces over compacta, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3433–3439. [7] Eric K. van Douwen and Washek F. Pfeffer, Some properties of the Sor- genfrey line and related spaces, Pacific J. Math. 81 (1979), no. 2, 371–377. [8] Todd Eisworth, On 퐷-spaces, in Open Problems in Topology, II. Ed. Elliott Pearl. Amsterdam-Oxford: Elsevier, B.V., 2007. 129–134 [9] Gary Gruenhage, A note on 퐷-spaces, Topology Appl. 153 (2006), no. 13, 2229–2240. [10] Klaas Pieter Hart, Jun-iti Nagata, and Jerry E. Vaughan, Eds. Encyclo- pedia of General Topology. Amsterdam: Elsevier Science Publishers, B.V., 2004. [11] Witold Hurewicz, Uber¨ eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1926), 401–421. [12] I. Juhászand W. Weiss, Omitting the cardinality of the continuum in scattered spaces, Topology Appl. 31 (1989), no. 1, 19–27. [13] LúciaR. Junqueira and Franklin D. Tall, More reflections on compactness, Fund. Math. 176 (2003), no. 2, 127–141. [14] Janusz Pawlikowski, Undetermined sets of point-open games, Fund. Math. 144 (1994), no. 3, 279–285. 122 L. F. AURICHI

[15] Liang-XuePeng, On linear neighborhood assignments and dually discrete spaces, Topology Appl. 155 (2008), no. 16, 1867–1874. [16] Marion Scheepers, Selection principles and covering properties in topology, Note Mat. 22 (2003/04), no. 2, 3–41.

[17] Franklin D. Tall, On the cardinality of Lindel¨ofspaces with points 퐺훿, Topology Appl. 63 (1995), no. 1, 21–38.

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