aa r X i v : . [ m a t h . GN ] M a y SELECTIVE GAMES ON BINARY RELATIONS
RODRIGO R. DIAS ‡ AND MARION SCHEEPERS
Abstract.
We present a unified approach, based on dominating families inbinary relations, for the study of topological properties defined in terms ofselection principles and the games associated to them.
Introduction
Classical games introduced by Berner and Juhász [12], Galvin [18], Gruenhage[21] and Telgársky [35, 36] associated with diverse topological properties, as well asseveral subsequent games, can be considered in a single unifying framework basedon the notion of a relation , defined below in Definition 2.1, and of a dominatingfamily for a relation, defined below in Definition 2.2. We define these games inDefinition 3.3. This framework subsumes and clarifies several isolated theoremsabout topological games. One of the two main phenomena about these topological games that is addressedhere is: Consider a topological property, E . For many examples of E one can findspaces X and Y each of which has the property E , while the product space X × Y does not have the property E . Define a space X to be productively E if, for eachspace Y that has property E , also X × Y has the property E . For some properties E it is a significant mathematical problem to characterize the spaces X that areproductively E . For several isolated examples of topological games G it has beenfound that, if a certain player of the game G on a space X has a winning strategy,then X is productively E .We study the productivity of properties E in this abstract context. The fourproperties E we consider — which will be described in detail after Definition 3.9 —are as follows: E : TWO has a winning strategy in the game G ; E : ONE does not have a winning strategy in the game G ; E : A selective version of a certain countability hypothesis holds; E : A certain countability hypothesis holds.For the properties we consider it will be the case that E ⇒ E ⇒ E ⇒ E , where sometimes, but not always, an implication is reversible.The nature of our theorems is as follows: Mathematics Subject Classification.
Primary 91A44; Secondary 54B10, 54D20, 54D45,54D65, 54D99.
Key words and phrases. topological games, selection principles, binary relations, productspaces. ‡ Supported by FAPESP (2013/10363-3). Which specific theorems in the literature are affected will be presented later in the paper. · A relation in the class E is productively E . · A relation in the class E is productively E . · A relation in the class E is productively E .It is curious that as of yet the techniques for the other implications do not seemto produce the implication that a space in the class E is productively E . Somequestions related to this issue will be posed at the end of the paper.These results about products bring us to analyzing situations in which TWO hasa winning strategy in a game, and the second of the two main phenomena addressedhere: In many instances of games G it is known that, if player TWO has a winningstrategy in the game G , then player TWO has a winning strategy in a game G ′ , inwhich the winning condition for TWO appears more stringent.This paper is organized as follows: After establishing notational conventions inSection 1 we introduce a general framework for the theory regarding E in Section 2.In Section 3 we translate classical duality results on games to the new framework.In Sections 4 and 5 we prove product theorems. Sections 6 and 7 are dedicated tosituations in which the existence of a winning strategy for player TWO in a certaingame turns out to be equivalent to the same condition in other games that areseemingly more difficult for TWO. In Section 8 we study conditions under which aLindelöf-like property is equivalent to ONE not having a winning strategy in theselective game associated to the property being considered. Section 9 contains somefinal remarks about the results presented in the paper.1. Notational conventions
Throughout our paper X denotes the underlying set of a topological space and τ denotes the ambient topology on the space. Whenever a second topological space Y is involved, its topology will be denoted by ρ . Unless explicitly stated otherwise,we do not make any assumptions about separation hypotheses on the topologicalspaces in our results.For a space X and a point x ∈ X , we write τ x = { U ∈ τ : x ∈ U } . For a subset A of X , the closure of A in X is denoted by A . The set of all compact subsets of X is denoted by K ( X ) .A set A is countable if | A | ≤ ℵ . For a cardinal number λ , we write [ A ] <λ = { B ⊆ A : | B | < λ } , [ A ] λ = { B ⊆ A : | B | = λ } and [ A ] ≤ λ = [ A ] <λ ∪ [ A ] λ . The setof all functions from a set A to a set B is denoted by A B ; we also write <ω B for S n ∈ ω n B .Throughout the paper we will make use of several families associated to a topo-logical space X ; the reader is referred to Example 2.3 for both their definition andthe notation we adopt to denote these families.For definitions of concepts found in the paper that are neither listed here nordefined right before the result in which they appear, the reader is referred to [16],[23] and [26].We end this section with two definitions we shall make use of frequently in thepaper. Definition 1.1.
Let A and B be families of sets. We say that ( A , B ) -Lindelöf holdsif every element of A has a countable subset that is an element of B . If A = B , wewill say A -Lindelöf instead of ( A , B ) -Lindelöf. Definition 1.2.
Let A and B be families of sets. ELECTIVE GAMES ON BINARY RELATIONS 3 · The notation S ( A , B ) abbreviates the following statement:For every sequence ( A n ) n ∈ ω of elements of A , there is a sequence ( B n ) n ∈ ω such that B n ∈ A n for all n ∈ ω and { B n : n ∈ ω } ∈ B . · The notation S fin ( A , B ) abbreviates the following statement:For every sequence ( A n ) n ∈ ω of elements of A , there is a sequence ( F n ) n ∈ ω such that F n ∈ [ A n ] < ℵ for all n ∈ ω and S n ∈ ω F n ∈ B . Note that S ( A , B ) implies S fin ( A , B ) , which in turn implies ( A , B ) -Lindelöf.2. Relations and dominating families
All of our main results in this paper are phrased in terms of dominating familiesin binary relations , a general framework that allows us to express a number oftopological concepts through a unified terminology. This section is dedicated tostating the basic definitions and exploring how this framework can be used tocapture properties of interest in general topology.The following definition is based on [42]; see also [13, Section 4].
Definition 2.1. A relation is a triple ( A, B, R ) , where A = ∅ and R ⊆ A × B issuch that ∀ a ∈ A ∃ b ∈ B (( a, b ) ∈ R ) . We will henceforth adopt the convention of writing aRb instead of ( a, b ) ∈ R ,and we will read this as “ b dominates a in R ”. Definition 2.2.
For a relation P = ( A, B, R ) , define Dom( P ) = { Z ⊆ B : ∀ a ∈ A ∃ b ∈ Z ( aRb ) } . The elements of
Dom( P ) are said to be dominating in P . The table in Example 2.3 has double purpose: It serves to illustrate instancesof relations whose set of dominating families have topological meaning, and also todefine the concepts and terminology that are listed therein.
Example 2.3.
Let X be a topological space with topology τ , and let x ∈ X . Thefollowing table lists several examples of families associated to X (and x ) that canbe expressed as the set Dom( P ) for some relation P . R. R. DIAS AND M. SCHEEPERS P Dom( P )( X, τ, ∈ ) O X = {U ⊆ τ : X = S U} ([ X ] < ℵ , τ, ⊆ ) Ω X = {U ⊆ τ : U is an ω -cover of X } ( K ( X ) , τ, ⊆ ) K X = {U ⊆ τ : U is a k -cover of X } ( K ( X ) , K ( X ) \ {∅} , R ) ,where HRK ↔ H ∩ K = ∅ M X = {M ⊆ K ( X ) \ {∅} : M is a moving-off family } ( τ \ {∅} , X, ∋ ) D X = { D ⊆ X : X = D } ( τ \ {∅} , τ, R ) ,where U RV ↔ U ∩ V = ∅ D X = {U ⊆ τ : X = S U} ( τ \ {∅} , τ \ {∅} , ⊇ ) Π X = {V ⊆ τ \ {∅} : V is a π -base for X } ( { ( a, U ) : a ∈ U ∈ τ } , τ, R ) ,where ( a, U ) RV ↔ a ∈ V ⊆ U B X = {B ⊆ τ : B is a base for X } ( τ x , τ x , ⊇ ) V x = {V ⊆ τ x : V is a local base for X at x } ( X \ { x } , τ x , / ∈ ) (assuming ( X, τ ) is T ) Ψ x = {V ⊆ τ x : V 6 = ∅ and T V = { x }} ( τ x , X, ∋ ) Ω x = { A ⊆ X : x ∈ A } ( τ x , [ X ] < ℵ \ {∅} , ⊇ ) π N x = {S ⊆ [ X ] < ℵ \ {∅} : S is a π -network at x } ( τ x , [ X ] ≤ℵ \ {∅} , ⊇ ) π N ℵ x = {S ⊆ [ X ] ≤ℵ \ {∅} : S is a π -network at x } Note that several classical topological properties can be phrased in terms offamilies of the form
Dom( P ) : For example, a topological space X is Lindelöf if andonly if O X -Lindelöf holds, and X is countably tight at a point x ∈ X if and only if Ω x -Lindelöf holds. 3. Games and duality
We now proceed to defining the basic games we shall consider in this paper.
Definition 3.1.
We say that two games G and G ′ are equivalent if both of thefollowing hold: · ONE has a winning strategy in the game G if, and only if, ONE has awinning strategy in G ′ ; and · TWO has a winning strategy in the game G if, and only if, TWO has awinning strategy in G ′ . Definition 3.2.
We say that two games G and G ′ are dual games if both of thefollowing hold: · ONE has a winning strategy in the game G if, and only if, TWO has awinning strategy in G ′ ; and · TWO has a winning strategy in the game G if, and only if, ONE has awinning strategy in G ′ . The first game we consider is a natural game associated with a pair of relations:
ELECTIVE GAMES ON BINARY RELATIONS 5
Definition 3.3.
Let P = ( A, B, R ) and Q = ( C, D, T ) be relations. The game G ( P, Q ) is defined as follows: In each inning n ∈ ω , player ONE chooses a n ∈ A ,and then player TWO chooses b n ∈ B with a n Rb n . ONE wins a play a , b , · · · , a n , b n · · · if { b n : n ∈ ω } ∈ Dom( Q ) . Otherwise, TWO wins. As the following examples show, there are several topological games studied inthe literature that can be regarded as instances of the game G ( P, Q ) introducedabove. Example 3.4.
Let P be the relation ( X, τ, ∈ ) . Then Dom( P ) is O X , the collectionof open covers of X . In this instance the game G ( P, P ) corresponds to the point-open game G ( X ) of Galvin, introduced in [18] . Example 3.5.
Let P be the relation ( τ \ {∅} , X, ∋ ) . Then Dom( P ) is D X , thecollection of dense subsets of X . In this instance the game G ( P, P ) corresponds tothe point-picking game G Dω ( X ) of Berner and Juhász, introduced in [12] . Example 3.6.
Fix a point x ∈ X . Let P be the relation ( τ x , X, ∋ ) . Then Dom( P ) is Ω x , the collection of subsets of X that have x as a cluster point. In this instancethe game G ( P, P ) corresponds to the game G cO, P ( X, x ) of Gruenhage, introduced in [21] . Example 3.7.
Let P be the relation ( X, τ, ∈ ) and Q be the relation ( τ \ {∅} , τ, T ) ,where U T V ↔ U ∩ V = ∅ . Then Dom( Q ) is D X , the collection of open familieswith union dense in X . In this instance the game G ( P, Q ) corresponds to the game θ ( X ) of Tkachuk, introduced in [39] . Now, towards the duality theorem mentioned in the introduction, define thefollowing game:
Definition 3.8.
Let A and B be families of sets with A 6 = ∅ . The game G ( A , B ) is played as follows. In each inning n ∈ ω , ONE first chooses a set A n ∈ A , andthen TWO responds with a B n ∈ A n . A play A , B , · · · , A n , B n , · · · is won by TWO if { B n : n ∈ ω } is an element of B . Otherwise, ONE wins. We shall also refer to the following variation of the previous game later on.
Definition 3.9.
Let A and B be families of sets with A 6 = ∅ . The game G fin ( A , B ) is played as follows. In each inning n ∈ ω , ONE first chooses a set A n ∈ A , andthen TWO responds with a finite subset F n of A n . A play A , F , · · · , A n , F n , · · · is won by TWO if S n ∈ ω F n is an element of B . Otherwise, ONE wins. It is immediate that, if TWO has a winning strategy in G ( A , B ) , then TWOhas a winning strategy in G fin ( A , B ) . Similarly, if ONE has a winning strategy in G fin ( A , B ) , then ONE has a winning strategy in G ( A , B ) .Furthermore, we have the following chain of implications (now defining in moredetail the four properties E – E considered in the Introduction): R. R. DIAS AND M. SCHEEPERS
TWO has a winning strategy in G ( A , B ) ⇓ ONE does not have a winning strategy in G ( A , B ) ⇓ S ( A , B ) ⇓ ( A , B ) -Lindelöf.Note that the same implications hold when the subscript “ ” is replaced with “ fin ”.The following duality result is a rephrasing of Theorem 1 of [18]. Theorem 3.10 (Galvin [18]) . For all relations P and Q , the games G ( P, Q ) and G (Dom( P ) , Dom( Q )) are dual.Proof. We will prove that, if TWO has a winning strategy in G (Dom( P ) , Dom( Q )) ,then ONE has a winning strategy in G ( P, Q ) . The reader shall find no difficulty inverifying the remaining three implications.The core of the proof is the following observation: If σ : ( <ω Dom( P )) \ {∅} → B is a strategy for TWO in G (Dom( P ) , Dom( Q )) , then( † ) ∀ s ∈ <ω Dom( P ) ∃ a s ∈ A ( { b ∈ B : a s Rb } ⊆ { σ ( s ⌢ ( Z )) : Z ∈ Dom( P ) } ) . Suppose, in order to get a contradiction, that there is s ∈ <ω Dom( P ) witnessingthe failure of ( † ) . Then, for each a ∈ A , there is b a ∈ B with aRb a such that thereis no Z ∈ Dom( P ) satisfying b a = σ ( s ⌢ ( Z )) . But then we get a contradiction fromthe fact that e Z = { b a : a ∈ A } ∈ Dom( P ) , since this implies that σ ( s ⌢ ( e Z )) = b a for some a ∈ A .Having proved ( † ) , let now σ : ( <ω Dom( P )) \ {∅} → B be a winning strategy forTWO in G (Dom( P ) , Dom( Q )) . A winning strategy for ONE in G ( P, Q ) can thenbe defined as follows.ONE’s initial move is a ∅ ∈ A . If b ∈ B is TWO’s response, it follows from ( † ) that there is Z ∈ Dom( P ) such that b = σ (( Z )) ; let then a ( Z ) ∈ A be ONE’snext move in the play. TWO will respond with some b ∈ B satisfying a ( Z ) Rb ;by ( † ) , there is Z ∈ Dom( P ) satisfying b = σ (( Z , Z )) ; ONE’s next move willthen be a ( Z ,Z ) ∈ A ; and so forth.By proceeding in this fashion, we obtain a sequence ( Z n ) n ∈ ω of elements of Dom( P ) and a play of G ( P, Q ) in which TWO’s move in the inning n ∈ ω is b n = σ (( Z k ) k ≤ n ) . Since σ is a winning strategy for TWO in G (Dom( P ) , Dom( Q )) ,it follows that { b n : n ∈ ω } = { σ (( Z k ) k ≤ n ) : n ∈ ω } ∈ Dom( Q ) ; therefore, thestrategy above described is a winning strategy for ONE in G ( P, Q ) . (cid:3) Theorem 3.10 expresses in terms of relations and dominating families the under-lying argument in duality results such as the following corollaries.The first one, in which the argument above was first presented, is Theorem 1of [18]. Recall that the
Rothberger game on X [18] is the game G ( O X , O X ) . Atopological space X is a Rothberger space [30] if S ( O X , O X ) holds. Corollary 3.11 (Galvin [18]) . For every topological space X , the point-open gameon X and the Rothberger game on X are dual games.Proof. Apply Theorem 3.10 with P = Q = ( X, τ, ∈ ) . (cid:3) The next corollary is the combination of Theorems 7 and 8 of [33].
ELECTIVE GAMES ON BINARY RELATIONS 7
Corollary 3.12 (Scheepers [33]) . For every topological space X , the games G Dω ( X ) and G ( D X , D X ) are dual games.Proof. Apply Theorem 3.10 with P = Q = ( τ \ {∅} , X, ∋ ) . (cid:3) The next corollary is Theorem 3.3(2) of [39].
Corollary 3.13 (Tkachuk [39]) . For a topological space X , the games θ ( X ) and G ( O X , D X ) are dual games.Proof. Apply Theorem 3.10 with P = ( X, τ, ∈ ) and Q = ( τ \ {∅} , τ, T ) , where U T V ↔ U ∩ V = ∅ . (cid:3) Corollary 3.14 (folklore) . For a topological space X and a point x ∈ X , the games G cO, P ( X, x ) and G (Ω x , Ω x ) on X are dual games.Proof. Apply Theorem 3.10 with P = Q = ( τ x , X, ∋ ) . (cid:3) Remark 3.15.
It is worth pointing out that, for two relations P = ( A, B, R ) and Q = ( C, D, T ) , the games G ( P, Q ) and G (Dom( P ) , Dom( Q )) are of interest only if B ∈ Dom( Q ) , for otherwise TWO (resp. ONE) wins every play of G ( P, Q ) (resp. G (Dom( P ) , Dom( Q )) ) trivially. Although we do not include this condition in thedefinition of these games (since it is not needed for most of our results), the readershould note that, in all of the topological situations we consider in this paper, thesets B and D are the same. Products and singleton selections
In this section we explore the behavior of some selective properties under prod-ucts, focusing on the game G ( P, Q ) — and, equivalently, on the game G (Dom( P ) , Dom( Q )) .The first step towards this goal is to define the product of two relations. Sincewe are interested in applying the general results to topological properties, we mustconsider a definition that allows us to do the following in as many situations aspossible: If P and Q are relations whose sets of dominating families correspond toa certain topological concept on the topological spaces X and Y respectively, thenthe set of dominating families in their product P ⊗ Q must allow us to describe thesame topological concept in the product space X × Y . A natural way of definingsuch product is the following. Definition 4.1.
The product relation of two relations P = ( A, B, R ) and P ′ =( A ′ , B ′ , R ′ ) is defined as P ⊗ P ′ = ( A × A ′ , B × B ′ , R ⊗ R ′ ) , where R ⊗ R ′ = { (( a, a ′ ) , ( b, b ′ )) ∈ ( A × A ′ ) × ( B × B ′ ) : aRb and a ′ R ′ b ′ } . Example 4.2.
Let P = ( τ \ {∅} , X, ∋ ) and Q = ( ρ \ {∅} , Y, ∋ ) . Then P ⊗ Q =(( τ \ {∅} ) × ( ρ \ {∅} ) , X × Y, ∋ ⊗ ∋ ) , where ( U, V ) ∋ ⊗ ∋ ( x, y ) ↔ ( U ∋ x & V ∋ y ) .In this case, we have Dom( P ) = D X , Dom( Q ) = D Y and Dom( P ⊗ Q ) = D X × Y . Example 4.3.
Let P = ( X, τ, ∈ ) and Q = ( Y, ρ, ∈ ) . Then P ⊗ Q = ( X × Y, τ × ρ, ∈⊗ ∈ ) , where ( x, y ) ∈ ⊗ ∈ ( U, V ) ↔ ( x ∈ U & y ∋ V ) . In this case, we have Dom( P ) = O X and Dom( Q ) = O Y . The set Dom( P ⊗ Q ) does not correspondexactly to O X × Y , but rather to the set of covers of the product X × Y constituted by basic open sets — which, however, is enough to express properties such as “ X × Y is Lindelöf ” and “ X × Y is Rothberger” as Dom( P ⊗ Q ) -Lindelöf and S (Dom( P ⊗ Q ) , Dom( P ⊗ Q )) respectively. R. R. DIAS AND M. SCHEEPERS
We can now prove the main result of this section — which was described ingeneral terms in the Introduction.
Proposition 4.4.
Let P , P ′ , Q and Q ′ be relations. Suppose that ONE has awinning strategy in the game G ( P ′ , Q ′ ) . ( a ) If (Dom( P ) , Dom( Q )) -Lindelöf holds, then (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) -Lindelöf also holds. ( b ) If S (Dom( P ) , Dom( Q )) holds, then S (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) alsoholds. ( c ) If ONE has a winning strategy in the game G ( P, Q ) , then ONE has a win-ning strategy in the game G ( P ⊗ P ′ , Q ⊗ Q ′ ) .Proof. Write P = ( A, B, R ) , P ′ = ( A ′ , B ′ , R ′ ) , Q = ( C, D, T ) and Q ′ = ( C ′ , D ′ , T ′ ) ,and let σ : <ω B ′ → A ′ be a winning strategy for ONE in the game G ( P ′ , Q ′ ) . ( a ) Let { ( b i , b ′ i ) : i ∈ I } ∈ Dom( P ⊗ P ′ ) be fixed. We will construct indexedfamilies h a ′ s : s ∈ <ω ω i and h i sn : s ∈ <ω ω, n ∈ ω i satisfying: · a ′ s ∈ A ′ for all s ∈ <ω ω ; · i sn ∈ I for all s ∈ <ω ω and n ∈ ω ; · { b i sn : n ∈ ω } ∈ Dom( Q ) for all s ∈ <ω ω ; and · for each f ∈ ω ω , (cid:18) a ′∅ , b ′ i ∅ f (0) , a ′ ( f (0)) , b ′ i ( f (0)) f (1) , a ′ ( f (0) ,f (1)) , b ′ i ( f (0) ,f (1)) f (2) , . . . , a ′ f ↾ k , b ′ i f ↾ kf ( k ) , . . . (cid:19) is a play of G ( P ′ , Q ′ ) in which ONE follows the strategy σ .We proceed by recursion. Suppose that k ∈ ω is such that h a ′ t : t ∈ Since n b m sk i : i ∈ I m sk and a ′ s R ′ b ′ m sk i o ∈ Dom( P ) for each k ∈ ω , we may apply S (Dom( P ) , Dom( Q )) to obtain a sequence ( i sk ) k ∈ ω satisfying: · i sk ∈ I m sk for each k ∈ ω ; · a ′ s R ′ b ′ m sk i sk for each k ∈ ω ; and · n b m sk i sk : k ∈ ω o ∈ Dom( Q ) .By recursion, this concludes the definition of i sk for s ∈ <ω ω and k ∈ ω . (In eachstep, what we have done is: if s = ( k , k , . . . , k h − ) ∈ h ω , then a ′ s is ONE’s movein a play of G ( P ′ , Q ′ ) whose history so far is a ′∅ , b ′ m ∅ k i ∅ k , a ′ ( k ) , b ′ m ( k k i ( k k , a ′ ( k ,k ) , b ′ m ( k ,k k i ( k ,k k , . . . , a ′ ( k ,k ,...,k h − ) , b ′ m ( k ,k ,...,kh − kh − i ( k ,k ,...,kh − kh − ! ; then, in view of the fact that Z n ( a ′ ) = { b ni : i ∈ I n and a ′ R ′ b ′ ni } is an element of Dom( P ) for all a ′ ∈ A ′ and n ∈ ω , we have made use of S (Dom( P ) , Dom( P ′ )) toselect from each I m sk with k ∈ ω an element b m sk i sk of Z m sk ( a ′ s ) in such a way that n b m sk i sk : k ∈ ω o ∈ Dom( Q ) .)We now claim that n(cid:16) b m sk i sk , b ′ m sk i sk (cid:17) : s ∈ <ω ω, k ∈ ω o ∈ Dom( Q ⊗ Q ′ ) . Indeed, let ( c, c ′ ) ∈ C × C ′ be arbitrary. Since n b m sk i sk : k ∈ ω o ∈ Dom( Q ) for every s ∈ <ω ω ,we may recursively pick, for each r ∈ ω , a k r ∈ ω such that cT b m ( kj ) j Proposition 4.5. Let P , P ′ , Q and Q ′ be relations. Suppose that TWO has awinning strategy in the game G (Dom( P ′ ) , Dom( Q ′ )) . ( a ) If (Dom( P ) , Dom( Q )) -Lindelöf holds, then (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) -Lindelöf also holds. ( b ) If S (Dom( P ) , Dom( Q )) holds, then S (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) alsoholds. ( c ) If TWO has a winning strategy in the game G (Dom( P ) , Dom( Q )) , thenTWO has a winning strategy in the game G (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) . We now present some instances of the previous propositions. The following listof results is not meant to exhaust the consequences that can be obtained fromPropositions 4.4 and 4.5, but rather to illustrate some of the contexts to whichthey can be applied. Corollary 4.6. If TWO has a winning strategy in the Rothberger game on a topo-logical space X , then X is productively Rothberger.Proof. Let Y be a Rothberger space. Now apply Proposition 4.5 ( b ) with P ′ = Q ′ =( X, τ, ∈ ) and P = Q = ( Y, ρ, ∈ ) . (cid:3) It is worth comparing Corollary 4.6 with Theorem 11(3) of [9], in which a productof two metric spaces is proven to be Rothberger under a weaker hypothesis on oneof the spaces and a stronger hypothesis on the other.The next result was first stated in [37]; see also Theorem 3.1 of [43]. Corollary 4.7 (Telgársky [37]) . The property “ONE has a winning strategy in thepoint-open game” (equivalently, “TWO has a winning strategy in the Rothbergergame”) is preserved under finite products.Proof. Let X and Y be spaces on which ONE has a winning strategy in the point-open game. Apply Proposition 4.4 ( c ) with P = Q = ( X, τ, ∈ ) and P ′ = Q ′ =( Y, ρ, ∈ ) . (cid:3) We thank Piotr Szewczak for bringing the paper [43] to our attention. ELECTIVE GAMES ON BINARY RELATIONS 11 The next result is also a consequence of Theorem 3.1 of [43]. By the compact-open game on a topological space X we mean the following game: In each inning n ∈ ω , ONE chooses a compact subset C n of X , and then TWO picks an open set U n with C n ⊆ U n ; ONE wins if X = S n ∈ ω U n , and loses otherwise. Corollary 4.8 (Yajima [43]) . The property “ONE has a winning strategy in thecompact-open game” is preserved under finite products.Proof. Let X and Y be spaces on which ONE has a winning strategy in the compact-open game. Apply Proposition 4.4 ( c ) with P = ( K ( X ) , τ, ⊆ ) , Q = ( X, τ, ∈ ) , P ′ =( K ( Y ) , ρ, ⊆ ) and Q ′ = ( Y, ρ, ∈ ) . The result follows from the observation that,whenever C ∈ K ( X ) , C ∈ K ( Y ) and W is an open subset of X × Y with C × C ⊆ W , there exist U ∈ τ and U ∈ ρ satisfying C × C ⊆ U × U ⊆ W . (cid:3) Corollary 4.9. Let X be a topological space on which TWO has a winning strategyin the game G (Ω , Ω) . Then: ( a ) the product X × Y satisfies S (Ω , Ω) for every topological space Y satisfying S (Ω , Ω) ; ( a ′ ) if Y is a topological space that is Rothberger in every finite power, then theproduct X × Y is Rothberger in every finite power; ( b ) if TWO has a winning strategy in the game G (Ω , Ω) on a topological space Y , then TWO has a winning strategy in G (Ω , Ω) on the product X × Y .Proof. By applying Proposition 4.5 ( b ) - ( c ) with P ′ = Q ′ = ([ X ] < ℵ , τ, ⊆ ) and P = Q = ([ Y ] < ℵ , ρ, ⊆ ) , we obtain ( a ) and ( b ) — note that every ω -cover U of theproduct X × Y has an open refinement V that is also an ω -cover and such thatevery element of V is a basic open set of the form U × V . Now ( a ′ ) follows from thefact that a topological space satisfies S (Ω , Ω) if and only if all of its finite powersare Rothberger [31]. (cid:3) For the next result, recall that a space X is weakly Lindelöf [17] if ( O X , D X ) -Lindelöf holds. Furthermore, X is weakly Rothberger [14] if S ( O X , D X ) holds. Corollary 4.10. Let X be a topological space on which TWO has a winning strategyin the game G ( O , D ) . Then: ( a ) the topological product X × Y is weakly Lindelöf whenever Y is a weaklyLindelöf space; ( b ) the topological product X × Y is weakly Rothberger whenever Y is a weaklyRothberger space; ( c ) TWO has a winning strategy in the game G ( O , D ) on the product X × Y whenever TWO has a winning strategy in G ( O , D ) on the topological space Y . Proof. Apply Proposition 4.5 with P ′ = ( X, τ, ∈ ) , Q ′ = ( τ \ {∅} , τ, T ) , P = ( Y, ρ, ∈ ) and Q = ( ρ \ {∅} , ρ, T ′ ) , where T = { ( U, V ) ∈ ( τ \ {∅} ) × τ : U ∩ V = ∅} and T ′ = { ( U, V ) ∈ ( ρ \ {∅} ) × ρ : U ∩ V = ∅} . (cid:3) The next corollary gives us an application to a nontopological context — namely,variations on the countable chain condition for partial orders (see e.g. [26, ChapterIII] for further details on the concepts involved in this result). A result similar to Corollary 4.10 ( c ) was independently obtained in [6] under a strongerassumption. Corollary 4.11. Let P be a partial order and PD P = { W ⊆ P : W is predense in P } . Suppose that TWO has a winning strategy in the game G ( PD P , PD P ) . Then: ( a ) the partial order P × Q is c.c.c. for every c.c.c. partial order Q ; ( b ) the partial order P × Q satisfies S ( PD , PD ) for every partial order Q sat-isfying S ( PD , PD ) .Proof. Apply Proposition 4.5 ( a ) - ( b ) with P ′ = Q ′ = ( P , P , ) and P = Q =( Q , Q , ) . The result follows from the observation that a partial order is c.c.c.if and only if every element of PD has a countable subset that is also an elementof PD — since for every X ∈ PD there is a maximal antichain A ∈ PD such thatevery element of X has an extension in A . (cid:3) It is worth mentioning that Corollary 1.7 of [15] is equivalent — by Theorem4.1 of the same paper — to the statement that the property “TWO has a winningstrategy in G ( PD , PD ) ” is preserved under arbitrary products with finite support.The next result may be viewed as a topological counterpart of Corollary 4.11. Corollary 4.12. Let X be a topological space on which TWO has a winning strategyin the game G ( D , D ) . Then: ( a ) the product X × Y is c.c.c. for every c.c.c. topological space Y ; ( b ) the product X × Y satisfies S ( D , D ) for every topological space Y thatsatisfies S ( D , D ) .Proof. Apply Corollary 4.11 with P = ( τ \ {∅} , ⊆ ) and Q = ( ρ \ {∅} , ⊆ ) . (cid:3) Again by Corollary 1.7 of [15] (see comment after Corollary 4.11), the property“TWO has a winning strategy in G ( D , D ) ” is preserved in arbitrary (Tychonoff)products of topological spaces.Note that, as a consequence of Corollary 4.12 ( a ) , TWO does not have a winningstrategy in the game G ( D , D ) played on a Suslin line; therefore, the game G ( D , D ) is undetermined on Suslin lines — see [32, Theorem 14 and observation followingProblem 1].For the next result, we recall that a topological space is R -separable [33] if itsatisfies S ( D , D ) . For a topological space X , we define δ ( X ) = sup { d ( Z ) : Z ∈ D X } , where d ( Z ) = min {| D | : D ∈ D Z } + ℵ . Corollary 4.13. Let X be a topological space such that TWO has a winning strategyin the game G ( D , D ) on X . Then: ( a ) δ ( X × Y ) = ℵ for every topological space Y with δ ( Y ) = ℵ ; ( b ) X × Y is R -separable for every R -separable space Y ; ( c ) TWO has a winning strategy in the game G ( D , D ) on the product X × Y for every topological space Y on which TWO has a winning strategy in thegame G ( D , D ) .Proof. Apply Proposition 4.5 with P ′ = Q ′ = ( τ \ {∅} , X, ∋ ) and P = Q = ( ρ \{∅} , Y, ∋ ) . (cid:3) For our last result in this section, recall that a topological space X is countablytight at a point x ∈ X if Ω x -Lindelöf holds on X , and has countable strong fantightness at x [31] if S (Ω x , Ω x ) holds on X . Corollary 4.14. Let X be a topological space and x ∈ X . Suppose that TWO hasa winning strategy in the game G (Ω x , Ω x ) on X . Then: ELECTIVE GAMES ON BINARY RELATIONS 13 ( a ) if a topological space Y is countably tight at a point y ∈ Y , then X × Y iscountably tight at ( x, y ) ; ( b ) if a topological space Y has countable strong fan tightness at y ∈ Y , then theproduct space X × Y has countable strong fan tightness at the point ( x, y ) ; ( c ) if a topological space Y is such that TWO has a winning strategy in thegame G (Ω y , Ω y ) for a point y ∈ Y , then TWO has a winning strategy inthe game G (Ω ( x,y ) , Ω ( x,y ) ) on the product X × Y .Proof. Apply Proposition 4.5 ( b ) with P ′ = Q ′ = ( { U ∈ τ : x ∈ U } , X, ∋ ) and P = Q = ( { V ∈ ρ : y ∈ V } , Y, ∋ ) . (cid:3) We note that Corollary 4.14 ( a ) extends Corollary 2.4 of [4], in which the sameconclusion is obtained under the assumption that X is completely regular. Thisresult will be further improved in Corollary 7.6, in view of the fact that, if π N ℵ x -Lindelöf holds on X , then X is productively countably tight at x [11, Corollary2.3]. 5. Products and finite selections By Corollary 3 of [38], Corollary 4.8 is equivalent to the statement that theproperty “TWO has a winning strategy in the Menger game” is finitely productivein the realm of regular spaces. We will now give a direct proof of this fact withoutassuming any separation axioms, which will serve as motivation for Proposition 5.3. Proposition 5.1. Let X and Y be topological spaces such that TWO has a winningstrategy in both of the games G fin ( O X , O X ) and G fin ( O Y , O Y ) . Then TWO has awinning strategy in the game G fin ( O X × Y , O X × Y ) .Proof. Let ϕ : <ω O X \ {∅} → [ τ ] < ℵ \ {∅} and σ : <ω O Y \ {∅} → [ ρ ] < ℵ \ {∅} bewinning strategies for TWO in G fin ( O X , O X ) and G fin ( O Y , O Y ) respectively. Let P = S n ∈ ω ( n ω × n ω ) , and write ω = ˙ S { L ts : ( s, t ) ∈ P} with · | L ts | = ℵ for every ( s, t ) ∈ P ; and · min L ts > max(im( s )) for every ( s, t ) ∈ P \ { ( ∅ , ∅ ) } .Enumerate each set L ts as L ts = { l ts ( j ) : j ∈ ω } , where l ts ( j ) < l ts ( j + 1) for all j ∈ ω .In order to construct a winning strategy for TWO in the game G fin ( O X × Y , O X × Y ) ,we may assume that, in each inning n ∈ ω of this game, ONE plays an open cover U n of X × Y of the form U n = { U ni × V ni : i ∈ I n } . Given such an open cover,for each x ∈ X , define I xn = { i ∈ I n : x ∈ U ni } and V xn = { V ni : i ∈ I xn } ∈ O Y .In each inning n ∈ ω , we will make use of the strategy σ to assign to each x ∈ X a finite nonempty subset of V xn — that is, a finite nonempty subset F xn of I xn . Wewill then define W xn = T i ∈ F xn U ni for each x ∈ X , and then make use of the strat-egy ϕ to choose finitely many elements of the open cover W n = { W xn : x ∈ X } of X — that is, a finite subset { x nk : k ∈ ω } of X , here enumerated with infiniterepetition of all of the terms. TWO’s answer to U n will then be the finite set F n = S k ∈ ω n U ni × V ni : i ∈ F x nk n o . In order to make sure that this will define awinning strategy, when making use of σ we will consider not all of the previousinnings, but only those listed (in a sense that will become clear in the next para-graph) by the sequences s, t ∈ <ω ω such that n ∈ L ts ; also, when making use of We could put this proof together thanks to an idea due to Leandro Aurichi, to whom we arevery grateful. ϕ we will consider a play of G fin ( O X , O X ) whose history is given not by all of theinnings , , . . . , n − , but only those in L ts ∩ n .Now for the details of the procedure. Suppose that, in the inning n ∈ ω of G fin ( O X × Y , O X × Y ) , the play so far is ( U , F , U , F , . . . , U n ) , where each F m is ofthe form F m = S k ∈ ω n U mi × V mi : i ∈ F x mk m o , as described in the previous para-graph. Let ( s, t ) ∈ P and j ∈ ω be such that n = l ts ( j ) ∈ L ts . For each x ∈ X , let F xn ∈ [ I xn ] < ℵ \ {∅} be such that σ (cid:18) V x s ( r ) t ( r ) s ( r ) (cid:19) r ∈ dom( s ) ⌢ ( V xn ) ! = { V ni : i ∈ F xn } .We can now define the open neighborhood W xn = T i ∈ F xn U ni for each x ∈ X , andthen consider W n = { W xn : x ∈ X } ∈ O X . Now let { x nk : k ∈ ω } be a fi-nite subset of X , enumerated with infinite repetition of all of the terms, satis-fying ϕ (cid:16)(cid:0) W l ts ( h ) (cid:1) h ≤ j (cid:17) = n W x nk n : k ∈ ω o . TWO’s answer in the n -th inning ofthe play ( U , F , U , F , . . . , U n ) of G fin ( O X × Y , O X × Y ) is then the finite subset F n = S k ∈ ω n U ni × V ni : i ∈ F x nk n o of U n .Let us now prove that this defines a winning strategy for TWO in G fin ( O X × Y , O X × Y ) .Let ( x, y ) ∈ X × Y be arbitrary. Pick n ∈ L ∅∅ with x ∈ S k ∈ ω W x n k n — such an n must exist, for otherwise ONE TWO W l ∅∅ (0) ( W x l ∅∅ (0) k l ∅∅ (0) : k ∈ ω ) W l ∅∅ (1) ( W x l ∅∅ (1) k l ∅∅ (1) : k ∈ ω ) ... ... W l ∅∅ ( j ) ( W x l ∅∅ ( j ) k l ∅∅ ( j ) : k ∈ ω ) ... ...would be a play of G fin ( O X , O X ) in which TWO plays according to the winningstrategy ϕ and loses. Let then k ∈ ω be such that x ∈ W x n k n . Similarly, we canrecursively pick n r ∈ L ( k l ) l ONE TWO V x n k n (cid:26) V n i : i ∈ F x n k n (cid:27) V x n k n (cid:26) V n i : i ∈ F x n k n (cid:27) ... ... V x nrkr n r n V n r i : i ∈ F x nrkr n r o ... ...is a play of G fin ( O Y , O Y ) in which TWO follows the winning strategy σ ; therefore,there is h ∈ ω such that, for some i ′ ∈ F x nrkr n r , we have y ∈ V n r i ′ . Since x ∈ W x nrkr n r = T i ∈ F xnrkrnr U n r i ⊆ U n r i ′ also holds, it follows that ( x, y ) ∈ U n r i ′ × V n r i ′ . Thus, the setsplayed by TWO cover the product X × Y . (cid:3) In order to capture the main aspect needed to adapt the proof of Proposition5.1 to the general setting of relations, we define the following auxiliary concept. Definition 5.2. Let P = ( A, B, R ) be a relation and (cid:22) be a partial order on theset B . We say that (cid:22) is · downwards P -compatible if, for all a ∈ A and b , b ∈ B , ( aRb & aRb ) → ∃ ˜ b ∈ B ( aR ˜ b & ˜ b (cid:22) b & ˜ b (cid:22) b ); · upwards P -compatible if, for all a ∈ A and b , b ∈ B , ( aRb & b (cid:22) b ) → aRb . We can now state the main result of this section. Proposition 5.3. Let P = ( A, B, R ) , P ′ = ( A ′ , B ′ , R ′ ) , Q = ( C, D, T ) and Q ′ =( C ′ , D ′ , T ′ ) be relations with B = D such that there is a partial order (cid:22) on B thatis both downwards P -compatible and upwards Q -compatible. Suppose that TWO hasa winning strategy in the game G fin (Dom( P ′ ) , Dom( Q ′ )) . ( a ) If (Dom( P ) , Dom( Q )) -Lindelöf holds, then (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) -Lindelöf holds. ( b ) If S fin (Dom( P ) , Dom( Q )) holds, then S fin (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) alsoholds. ( c ) If TWO has a winning strategy in the game G fin (Dom( P ) , Dom( Q )) , thenTWO has a winning strategy in the game G fin (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) .Proof. Let σ : <ω Dom( P ′ ) \ {∅} → [ B ′ ] < ℵ \ {∅} be a winning strategy for TWOin the game G fin (Dom( P ′ ) , Dom( Q ′ )) . ( a ) Let { ( b i , b ′ i ) : i ∈ I } ∈ Dom( P ⊗ P ′ ) be fixed. For each x ∈ A , define Z x = { b ′ i : i ∈ I x } , where I x = { i ∈ I : xRb i } ; note that Z x ∈ Dom( P ′ ) . We will construct indexed families h F sx : s ∈ <ω ω, x ∈ A i and h a sn : s ∈ <ω ω, n ∈ ω i with · F sx ∈ [ I x ] < ℵ \ {∅} for all s ∈ <ω ω and x ∈ A ; and · a sn ∈ A for all s ∈ <ω ω and n ∈ ω .We proceed recursively as follows.Suppose that k ∈ ω is such that h a tn : t ∈ The following consequence of Proposition 5.3 was originally proven by Telgárskyunder the extra assumption that the space is regular; it can be obtained by puttingtogether Corollary 14.14 and Theorem 14.12 of [35] and Corollary 3 of [38]. Herewe follow the standard terminology and refer to the game G fin ( O , O ) as the Mengergame ; a topological space is Menger [24] if it satisfies S fin ( O , O ) . Corollary 5.4 (Telgársky [35, 38], for regular spaces) . If TWO has a winningstrategy in the Menger game on a topological space X , then X is both productivelyLindelöf and productively Menger.Proof. Let Y be a Lindelöf (respectively, Menger) space. Apply Proposition 5.3 ( a ) (respectively, ( b ) ) with P ′ = Q ′ = ( X, τ, ∈ ) , P = Q = ( Y, ρ, ∈ ) and U (cid:22) V ↔ U ⊆ V . (cid:3) As in the observation made after Corollary 4.6, it is worth comparing Corollary5.4 with Theorem 11(2) of [9].It is also worth remarking that, assuming the Continuum Hypothesis, one canconstruct a Sierpiński set S ⊆ R such that R \ Q is a continuous image of S × S [25,Theorem 2.12], which implies that S × S is not Menger. By Proposition 14 of [9],TWO has a winning strategy in the “length-( ω + ω )” variation of the Menger gameplayed on every Sierpiński set. This shows that the existence of a winning strategyfor TWO in this longer version of the Menger game, although stronger than theMenger property, is not strong enough to imply its productivity. Corollary 5.5. Let X be a topological space on which TWO has a winning strategyin the game G fin (Ω , Ω) . Then: ( a ) if Y is a topological space such that Ω Y -Lindelöf holds, then Ω X × Y -Lindelöfholds; ( a ′ ) if Y is a topological space such that every finite power of Y is Lindelöf, thenevery finite power of X × Y is Lindelöf; ( b ) the product X × Y satisfies S fin (Ω , Ω) for every topological space Y satisfying S fin (Ω , Ω) ; ( b ′ ) if Y is a topological space such that every finite power of Y is Menger, thenevery finite power of X × Y is Menger; ( c ) if TWO has a winning strategy in the game G fin (Ω , Ω) on a topological space Y , then he has a winning strategy in G fin (Ω , Ω) on the product X × Y .Proof. First, we recall that a topological space Y is Lindelöf in every finite powerif and only if Ω Y -Lindelöf holds [20], and that it is Menger in every finite power ifand only if S fin (Ω Y , Ω Y ) holds [25, Theorem 3.9]. Now apply Proposition 5.3 with P ′ = Q ′ = ([ X ] < ℵ , τ, ⊆ ) , P = Q = ([ Y ] < ℵ , ρ, ⊆ ) and U (cid:22) V ↔ U ⊆ V , having inmind the observation made in the proof of Corollary 4.9. (cid:3) For a topological space X , the space of all real-valued continuous functions on X regarded as a subspace of the (Tychonoff) power R X is denoted by C p ( X ) . Recallthat a topological space has countable fan tightness [3] at a point x if S fin (Ω x , Ω x ) holds. ELECTIVE GAMES ON BINARY RELATIONS 19 Corollary 5.6. Let X and Y be completely regular spaces. Assume that TWO hasa winning strategy in the game G fin (Ω , Ω ) played on C p ( X ) . ( a ) If C p ( Y ) is countably tight, then C p ( X ) × C p ( Y ) is also countably tight. ( b ) If C p ( Y ) has countable fan tightness, then the product C p ( X ) × C p ( Y ) alsohas countable fan tightness. ( c ) If TWO has a winning strategy in G fin (Ω , Ω ) played on C p ( Y ) , then TWOhas a winning strategy in G fin (Ω ( , ) , Ω ( , ) ) played on the product C p ( X ) × C p ( Y ) .Proof. This follows from Corollary 5.5, in view of the following results: · for a completely regular space X , TWO has a winning strategy in G fin (Ω , Ω) on X if and only if TWO has a winning strategy in G fin (Ω , Ω ) on C p ( X ) [34, Theorem 26]; · a completely regular space X is Menger in every finite power if and only if C p ( X ) has countable fan tightness [3, Theorem 4]; · a completely regular space X is Lindelöf in every finite power if and onlyif C p ( X ) is countably tight — see [2, Theorem 2] and [29, Theorem 1]. (cid:3) The hypothesis about the partial order (cid:22) in Proposition 5.3 is essential and thereis no possibility of obtaining a result as general as Proposition 4.5 for propertiesinvolving finite selections. This can be seen, for example, by considering the sameproperties appearing in Corollary 5.6: It follows from a result of Uspenski˘ı [41,Theorem 1] that a space X is Lindelöf in the G δ -topology is and only if C p ( X ) isproductively countably tight; note that, by Theorem 26 of [34], TWO has a winningstrategy in G fin (Ω , Ω ) on C p ( R ) since TWO has a winning strategy in G fin (Ω , Ω) on R [10, Lemma 2], and yet C p ( R ) is not productively countably tight in view ofUspenski˘ı’s result. 6. γ -dominating sequences In this section we study games defined in terms of γ -dominating sequences , whichallow us to express convergence-like properties.We start off with a simple fact: Lemma 6.1. Let P = ( A, B, R ) be a relation and ( z n ) n ∈ ω ∈ ω B . The followingassertions are equivalent: ( a ) { n ∈ ω : ¬ ( aRz n ) } is finite for all a ∈ A ; ( b ) { z n : n ∈ X } ∈ Dom( P ) for all X ∈ [ ω ] ℵ . Definition 6.2. Let P = ( A, B, R ) be a relation. A sequence ( z n ) n ∈ ω ∈ ω B is γ -dominating in P if the conditions in Lemma 6.1 hold. The family of all γ -dominating sequences in P will be denoted by Dom γ ( P ) . We shall now consider variations of the games G and G involving γ -dominatingsequences. Definition 6.3. Let P and Q be relations. ( a ) The game G γ ( P, Q ) is played according to the same rules as G ( P, Q ) , butnow ONE wins if ( b n ) n ∈ ω ∈ Dom γ ( Q ) , and loses otherwise. ( b ) In a slight abuse of notation (cf. Definition 3.8), we shall designate by G (Dom( P ) , Dom γ ( Q )) the game played according to the same rules as G (Dom( P ) , Dom( Q )) , but in which the winner is TWO if ( b n ) n ∈ ω ∈ Dom γ ( Q ) and ONE otherwise. The games defined above satisfy the following duality theorem, which parallelsTheorem 3.10. Theorem 6.4 (Galvin [18]) . The games G γ ( P, Q ) and G (Dom( P ) , Dom γ ( Q )) aredual for all relations P and Q . The following consequence of Theorem 6.4 was observed in the proof of Propo-sition 1 of [19]. First, let us recall the game G O,P ( X, x ) , introduced by Gruenhagein [21]. Let a topological space X and x ∈ X be fixed. In each inning n ∈ ω , ONEpicks V n ∈ τ x , and then TWO chooses x n ∈ V n . ONE wins if the sequence ( x n ) n ∈ ω converges to x , and loses otherwise. Corollary 6.5 (Gerlits [19]) . Let X be a topological space, x ∈ X and Γ x = { ( x n ) n ∈ ω ∈ ω X : x n n →∞ −→ x } . Then the game G O,P ( X, x ) and the game G (Ω x , Γ x ) on X are dual.Proof. Apply Theorem 6.4 with P = Q = ( τ x , X, ∋ ) . (cid:3) Our main goal in this section is to find conditions under which the existence ofa winning strategy for ONE in the game G ( P, Q ) yields the existence of a winningstrategy for ONE also in the game G γ ( P, Q ) . In order to formulate such conditions,we will need the following auxiliary notion. Definition 6.6 (cf. Definition 5.2) . Let P = ( A, B, R ) be a relation. We say thata partial order E on A is downwards P -compatible if, for all a , a ∈ A and b ∈ B , ( a E a & a Rb ) → a Rb. The argument for the next result is essentially taken from Theorem 3.9 of [21](which we state as Corollary 6.8); see also Theorem 1 of [20] (also stated here asCorollary 6.10). Theorem 6.7 (Gruenhage [21]) . Let P = ( A, B, R ) and Q = ( C, D, T ) be relations.Suppose that there is a downwards P -compatible partial order E on A such that,for each finite subset F of A , there is ˜ a ( F ) ∈ A satisfying a E ˜ a ( F ) for all a ∈ F .Then the following conditions are equivalent: ( a ) ONE has a winning strategy in G ( P, Q ) ; ( b ) ONE has a winning strategy in G γ ( P, Q ) .Proof. The implication ( b ) → ( a ) is immediate. We will prove that ( a ) implies ( b ) .Let σ : <ω B → A be a winning strategy for ONE in G ( P, Q ) . For each n ∈ ω ,let S n be the (finite) set of all strictly increasing sequences with range included in n . Now define ϕ : <ω B → A by ϕ (( b j ) j Proposition 6.9 (Telgársky [35]) . The point-open game is equivalent to the finite-open game, which is played according to the following rules: In each inning n ∈ ω ,ONE picks a finite subset F n of X , and then TWO chooses U n ∈ τ with F n ⊆ U n ;ONE wins if { U n : n ∈ ω } ∈ O X , and TWO wins otherwise. With Proposition 6.9 in mind, we recall the strict point-open game , introducedin [20]. The game is played according to the same rules as the finite-open game, but now ONE wins if X = S n ∈ ω T m ∈ ω \ n U m . Corollary 6.10 (Gerlits–Nagy [20]) . Let X be a topological space. The followingstatements are equivalent: ( a ) ONE has a winning strategy in the point-open game on X ; ( b ) ONE has a winning strategy in the strict point-open game on X .Proof. Apply Theorem 6.7 with P = ([ X ] < ℵ , τ, ⊆ ) , Q = ( X, τ, ∈ ) and F E G ↔ F ⊆ G . (cid:3) For the next two corollaries, we recall the game G ∗ ( X ) introduced by Gruenhagein [22] for every noncompact space X . In each inning n ∈ ω of G ∗ ( X ) , ONE picksa compact set C n ⊆ X , and then TWO picks a nonempty compact set L n ⊆ X with C n ∩ L n = ∅ . ONE wins if the family { L n : n ∈ ω } is locally finite, and losesotherwise. We shall also consider a variation G ∗∗ ( X ) of this game, which is playedaccording to the same rules but now ONE wins if and only if { L n : n ∈ ω } ∈ M X . Corollary 6.11. Let X be a noncompact locally compact space. The followingassertions are equivalent: ( a ) ONE has a winning strategy in G ∗ ( X ) ; ( b ) ONE has a winning strategy in G ∗∗ ( X ) .Proof. For ( b ) → ( a ) , let σ be a winning strategy for ONE in G ∗∗ ( X ) . We can definea strategy ϕ for ONE in G ∗ ( X ) by setting ϕ (( L i ) i TWO’s moves in a play in which ONE follows ϕ is an infinite locally finite familyof nonempty compact sets, and hence is a moving-off family by Lemma 4 of [8].For ( a ) → ( b ) , apply Theorem 6.7 with P = Q = ( K ( X ) , K ( X ) \ {∅} , R ) , where CRL ↔ C ∩ L = ∅ . The result follows from the observation that, if X is locallycompact and ( L n ) n ∈ ω ∈ Dom γ ( Q ) , then { L n : n ∈ ω } is locally finite. (cid:3) The next corollary presents some variations on the game-theoretic characteriza-tion of paracompactness for locally compact T spaces obtained by Gruenhage in[22] — which states that a locally compact T space is paracompact if and only ifONE has a winning strategy in the game G ∗ ( X ) . Corollary 6.12. Let X be a noncompact locally compact T space and L = {L ⊆ K ( X ) \ {∅} : L is locally finite } . The following conditions are equivalent: ( a ) X is paracompact; ( b ) ONE has a winning strategy in G ∗ ( X ) ; ( c ) ONE has a winning strategy in G ∗∗ ( X ) ; ( d ) TWO has a winning strategy in G ( M , L ) ; ( e ) TWO has a winning strategy in G ( M , M ) .Proof. We have just proven ( b ) ↔ ( c ) in Corollary 6.11. The equivalences ( a ) ↔ ( b ) and ( b ) ↔ ( d ) are Theorem 5 of [22] and Theorem 2 of [8] respectively. Finally, ( c ) ↔ ( e ) follows from Theorem 3.10 with P = Q = ( K ( X ) , K ( X ) \ {∅} , R ) , where CRL ↔ C ∩ L = ∅ . (cid:3) ℵ -modifications In this section, we study another variation of the game G ( P, Q ) (resp. G (Dom( P ) , Dom( Q )) )for which the existence of a winning strategy for player ONE (resp. TWO) althoughapparently stronger, turns out to be equivalent to the same condition for the originalgame.This variation will be defined in terms of the following concept. Definition 7.1. The ℵ -modification of a relation P = ( A, B, R ) is the relation P ℵ = ( A, [ B ] ≤ℵ \ {∅} , e R ) , where a e RE ↔ ∀ b ∈ E ( aRb ) . The equivalence previously mentioned can then be stated as follows. Proposition 7.2. Let P and Q be relations. The following conditions are equiva-lent: ( a ) ONE has a winning strategy in the game G ( P, Q ) ; ( b ) ONE has a winning strategy in the game G ( P ℵ , Q ℵ ) .Proof. Write P = ( A, B, R ) and Q = ( C, D, T ) .The implication ( b ) → ( a ) is immediate, since G ( P, Q ) is equivalent to the game G ( P ℵ , Q ℵ ) played with the additional restriction that TWO must choose one-element subsets of B .For ( a ) → ( b ) , let σ : <ω B → A be a winning strategy for ONE in G ( P, Q ) . Fixan injective function s m s from <ω ω to ω satisfying s ⊆ t → m s ≤ m t for every s, t ∈ <ω ω — for example, define m s = Q i ∈ dom( s ) p s ( i )+1 i , where p i is the i -th primenumber. Now write each E ∈ [ B ] ≤ℵ \ {∅} as E = { b Ek : k ∈ ω } , and let ˜ a ∈ A be ELECTIVE GAMES ON BINARY RELATIONS 23 fixed. Define ϕ : <ω ([ B ] ≤ℵ \ {∅} ) → A by ϕ (( E j ) j Corollary 7.3. Let P and Q be relations. The following conditions are equivalent: ( a ) TWO has a winning strategy in the game G (Dom( P ) , Dom( Q )) ; ( b ) TWO has a winning strategy in the game G (Dom( P ℵ ) , Dom( Q ℵ )) . The following result is Theorem 5.1 of [36]. Given a nonempty family K ofsubsets of a topological space X , we call K -open game on X the game in which,in each inning n ∈ ω , ONE chooses K n ∈ K and then TWO picks an open set U n ⊆ X with K n ⊆ U n ; the winner is ONE if X = S n ∈ ω U n , and TWO otherwise.The K - G δ game on X is played according to the same rules, replacing “open” with“ G δ ”. Corollary 7.4 (Telgársky [36]) . Let X be a topological space and K be a nonemptyfamily of subsets of X . The following conditions are equivalent: ( a ) ONE has a winning strategy in the K -open game on X ; ( b ) ONE has a winning strategy in the K - G δ game on X .Proof. Apply Proposition 7.2 with P = ( K , τ, ⊆ ) and Q = ( X, τ, ∈ ) . The resultfollows from the observation that the games K - G δ and G ( P ℵ , Q ℵ ) are equivalent. (cid:3) As another consequence of Proposition 7.2, we have: Corollary 7.5. Let X be a topological space and x ∈ X . The following conditionsare equivalent: ( a ) TWO has a winning strategy in the game G (Ω x , Ω x ) ; ( b ) TWO has a winning strategy in the game G ( π N x , π N x ) ; ( c ) TWO has a winning strategy in the game G ( π N ℵ x , π N ℵ x ) .Proof. It is clear that ( c ) → ( b ) → ( a ) . Now the equivalence between ( a ) and ( c ) follows from Corollary 7.3 with P = Q = ( τ x , X, ∋ ) . (cid:3) As an immediate consequence of Corollary 7.5 (see also Corollary 8.6), we havethe following result, which answers Question 4.9 of [4] in the affirmative. Corollary 7.6. Let X be a topological space and x ∈ X . If TWO has a winningstrategy in the game G (Ω x , Ω x ) on X , then π N ℵ x -Lindelöf holds. ℵ -preserving relations Inspired by some features of relations of the form P ℵ introduced in the previoussection (see Lemma 8.3), we will now aim at finding general conditions on relationsunder which a Lindelöf-like property turns out to be strong enough to yield thenonexistence of a winning strategy for ONE in the associated selective game. Definition 8.1. Let P = ( A, B, R ) be a relation and (cid:22) be a partial order on B .We say that (cid:22) is countably downwards P -compatible if, for every a ∈ A and every E ∈ [ B ] ≤ℵ \ {∅} , ( ∀ b ∈ E ( aRb )) → ∃ ˜ b ∈ B ( aR ˜ b & ∀ b ∈ E (˜ b (cid:22) b )) . Definition 8.2. A relation P = ( A, B, R ) is ℵ -preserving if there is a partialorder (cid:22) on B that is both upwards P -compatible (see Definition 5.2) and countablydownwards P -compatible. Our main examples of ℵ -preserving relations will be of the form P ℵ : Lemma 8.3. Let P be a relation. Then P ℵ (see Definition 7.1) is an ℵ -preservingrelation.Proof. Just note that, if P = ( A, B, R ) , then the partial order (cid:22) on [ B ] ≤ℵ definedby E (cid:22) E ↔ E ⊇ E witnesses the fact that P ℵ is ℵ -preserving. (cid:3) The following proposition is the main result of this section. Proposition 8.4. Let P be an ℵ -preserving relation. The following conditionsare equivalent: ( a ) Dom( P ) -Lindelöf; ( b ) S (Dom( P ) , Dom( P )) ; ( c ) ONE does not have a winning strategy in the game G (Dom( P ) , Dom( P )) . ELECTIVE GAMES ON BINARY RELATIONS 25 Proof. Clearly, ( c ) → ( b ) → ( a ) . We will prove the implication ( a ) → ( c ) .Let a strategy for ONE in G (Dom( P ) , Dom( P )) be fixed. By ( a ) , we mayassume that each of ONE’s moves in this strategy is a countable set; this allowsus to regard such strategy as an indexed family ( b t ) t ∈ <ω ω \{∅} — meaning that, if s ∈ n ω is such that TWO’s choices in the first n innings were ( b s ↾ i ) i The following statements are equivalent for a topological space X : ( a ) X is strongly Alster; ( b ) S ( G K , G K ) ; ( c ) ONE does not have a winning strategy in the game G ( G K , G K ) .Proof. Apply Proposition 8.4 with P = ( K ( X ) , G δ ( X ) , ⊆ ) , where K ( X ) = { C ⊆ X : C is compact } and G δ ( X ) = { W ⊆ X : W is a countable intersection of opensets } . (Note that W (cid:22) W ↔ W ⊆ W witnesses that P is ℵ -preserving.) (cid:3) The next corollary deals with a game that was also explored in Corollary 7.5. Corollary 8.6. Let X be a topological space and x ∈ X . The following conditionsare equivalent: ( a ) π N ℵ x -Lindelöf holds; ( b ) S ( π N ℵ x , π N ℵ x ) ; ( c ) ONE does not have a winning strategy in the game G ( π N ℵ x , π N ℵ x ) .Proof. Note that P = ( τ x , [ X ] ≤ℵ , ⊇ ) is ℵ -preserving by Lemma 8.3. Now applyProposition 8.4. (cid:3) We note that the equivalence between ( a ) and ( b ) in Corollary 8.6 also followsfrom Proposition 2.5(2) of [11]. 9. Remarks A. As not all topological properties can be expressed in terms of relations, it shouldbe made clear that there are selective topological games that have been studied inthe literature for which the analogue of the previous results does not hold. Weillustrate this with the following selective property: A topological space X is se-lectively screenable [1] if, for every sequence ( U n ) n ∈ ω of open covers of X , there is a sequence ( V n ) n ∈ ω of families of open subsets of X such that X = S n ∈ ω V n andeach V n is a pairwise disjoint partial refinement of U n . It follows from Example 1of [28] and Theorem 2.2 of [7] that TWO having a winning strategy in the gamenaturally associated with selective screenability does not imply that the space isproductively selectively screenable; therefore, a result similar to Propositions 4.5and 5.3 could not be obtained for this concept.B. Having in mind the properties E – E from the Introduction, there seems to bea gap in Propositions 4.5 and 5.3, which motivates the two main open questions ofthis paper: Problem 9.1. Let P , P ′ , Q and Q ′ be relations such that that TWO has a winningstrategy in the game G (Dom( P ′ ) , Dom( Q ′ )) and ONE does not have a winningstrategy in the game G (Dom( P ) , Dom( Q )) . Does it follow that ONE does not havea winning strategy in the game G (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) ? Problem 9.2. Let P = ( A, B, R ) , P ′ = ( A ′ , B ′ , R ′ ) , Q = ( C, D, T ) and Q ′ =( C ′ , D ′ , T ′ ) be relations with B = D such that: · there is a partial order (cid:22) on B that is both downwards P -compatible andupwards Q -compatible; · TWO has a winning strategy in the game G fin (Dom( P ′ ) , Dom( Q ′ )) ; and · ONE does not have a winning strategy in the game G fin (Dom( P ) , Dom( Q )) .Does it follow that ONE does not have a winning strategy in the game G fin (Dom( P ⊗ P ′ ) , Dom( Q ⊗ Q ′ )) ? It should be pointed out that, in many instances of the topological properties inwhich we are interested in this paper, it is the case that ONE does not have a win-ning strategy in the game G (Dom( P ) , Dom( Q )) if and only if S (Dom( P ) , Dom( Q )) holds (and similarly for G fin and S fin ); see e.g. [24, Theorem 10] (for the Mengergame), [27, Lemma 2] (for the Rothberger game) and [32, Theorems 2 and 14] (forthe games G fin ( D , D ) and G ( D , D ) ). As a consequence, none of these instancescould provide us with a negative answer to Problems 9.1 and 9.2.It is also known that there are other instances in which this equivalence doesnot hold — such as G ( D , D ) [33, Example 3] —, which could be a first attempt toanswer Problem 9.1 in the negative. More explicitly: Problem 9.3. Let X and Y be topological spaces such that TWO has a winningstrategy in the game G ( D X , D X ) and ONE does not have a winning strategy in thegame G ( D Y , D Y ) . Does it follow that ONE does not have a winning strategy inthe game G ( D X × Y , D X × Y ) ? C. Regarding the four properties E i mentioned in the Introduction, and having inmind Proposition 4.5, one could ask whether, for some pair ( i, j ) with ≤ i ≤ j ≤ ,it is the case that every relation in the class E i is productively E j . This possibilitycan be ruled out by considering the following.For a topological space X and P = Q = ( X, τ, ∈ ) , we have: E : ONE does not have a winning strategy in the game G ( O X , O X ) ; E : S ( O X , O X ) ; E : X is a Lindelöf space.By Lemma 2 of [27], E and E are equivalent. In Theorem 8 of [40], two Roth-berger spaces are constructed in such a way that their product is not Lindelöf. The ELECTIVE GAMES ON BINARY RELATIONS 27 conjunction of these results shows that E is not strong enough to imply produc-tivity with respect to either E , E or E . Acknowledgements This research was done during a visit of the first author to the Department ofMathematics at Boise State University. The author wishes to express his gratitudeto the Department for their hospitality and academic support.We are deeply indebted to Samuel Coskey, whose insightful comments led usto the approach of dominating families here presented. 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