On some topological games involving networks

In these notes we introduce and investigate two new games called R-nw-selective game and the M-nw-selective game. These games naturally arise from the corresponding selection principles involving networks introduced in \cite{BG}.


Introduction
Throughout the paper we mean by "space"a topological Hausdorff space.A family N of sets is called a network for X if for every x ∈ X and for every open neighbourhood U of x there exists an element N of N such that x ∈ N ⊆ U; nw(X) = min{|N | : N is a network for X} is the network weight of X.A family P of open sets is called a π-base for X if every nonempty open set in X contains a nonempty element of P; πw(X) = min{|P| : P is a π-base for X} is the π-weight of X.It is known that nw(X) ≤ w(X), where w(X) denotes the weight of the space X, and in the class of compact Hausdorff spaces nw(X) = w(X) (see [2]).δ(X) = sup{d(Y ) : Y is a dense subset of X}, where d(X) denotes the density of the space X.Recall that for f, g ∈ ω ω , f ≤ * g means that f (n) ≤ g(n) for all but finitely many n (and f ≤ g means that f (n) ≤ g(n) for all n ∈ ω).A subset D ⊆ ω ω is dominating if for each g ∈ ω ω there is f ∈ D such that g ≤ * f .The minimal cardinality of a dominating subset of ω ω is denoted by d.The value of d does not change if one considers the relation ≤ instead of ≤ * [9, Theorem 3.6].M denotes the family of all meager subsets of R. cov(M) is the minimum of the cardinalities of subfamilies U ⊆ M such that U = R.However, another description of the cardinal cov(M) is the following.cov(M) is the minimum cardinality of a family F ⊂ ω ω such that for every g ∈ ω ω there is f ∈ F such that f (n) = g(n) for all but finitely many n ∈ ω (see [3] and also [4,Theorem 2.4.1]).Thus if F ⊂ ω ω and |F | < cov(M), then there is g ∈ ω ω such that for every f ∈ F , f (n) = g(n) for infinitely many n ∈ ω; it is often said that g guesses F .Also if P is a countable poset and D is a family of dense sets of cardinality strictly less than cov(M) then there exists a generic that meets all the dense sets of the family [4, Section 3].
In [13,15] a systematic approach was considered to describe selection principles.Given two collections A and B of some particular topological objects on a space X, Scheepers introduced the following notation: S f in (A, B) : For every sequence (U n : n ∈ ω) of elements of A there exists a finite subset If one denotes by O the family of all open covers of a space X and by D the family of all dense subsets of a space X, a space is said to be Rothberger if it satisfies S 1 (O, O), Menger if it satisfies S f in (O, O), R-separable if it has the property S 1 (D, D) and M-separable if has the property S f in (D, D). δ(X) = ω for every M-separable space X [6] and if δ(X) = ω and πw(X) < d, then X is M-separable (a stronger version of this fact is estabilished in [15,Theorem 40]); moreover, if δ(X) = ω and πw(X) < cov(M), then X is Rseparable (a stronger version of this fact is estabilished in [15,Theorem 29]).Every space having a countable base is R-separable, therefore M-separable.However, not every space with countable network weight is M-separable.Hence in [7] the authors asked under which conditions a space with countable network weight must be M-separable and introduced and studied the following classes of spaces.Definition 1.1.Let X be a space with nw(X) = ω.
• X is M-nw-selective if for every sequence (N n : n ∈ ω) of countable networks for X one can select finite In [7] it was proved that any R-nw-selective (M-nw-selective) space is both Rothberger and R-separable (Menger and M-separable).See also [8] for more details about these two properties.Recall that topological games, introduced with a systematical structure in [13,15], are infinite games played by two different players, Alice and Bob, on a topological space X (see also [1]).We assume that the lenght (number of innings) of the games is ω and the two players pick in each inning some topological objects of a fixed space.The strategies of the two players are a priori defined; they are some functions that take care of the game history.At the end there is only one winner, so a draw is not allowed.Playing a game G on a space X gives rise to two properties: "Alice has a winning strategy in the game G on X"; "Bob has a winning strategy in the game G on X".Of course, since there is not draw, it is impossible for a space to have both these properties, but it can happen that the negation of both of them holds.In this case we say that the game G is undeterminate on the space X.Given two families of topological objects A and B, the followings are two games associated to selection principles.G 1 (A, B) : is played according to the following rules.
for every n ∈ ω Alice chooses A n ∈ A; G f in (A, B) : is played according to the following rules.
for every n ∈ ω Alice chooses A n ∈ A; -Bob answers picking a finite subset B n ⊆ A n for each n ∈ ω; The game G 1 (O, O), called Rothberger game, is strictly related to the Rothberger property.In the following we denote this game by Rothberger(X).The game G f in (O, O), called Menger game, is strictly related to the Menger property.In the following we denote this game by Menger(X).The game G 1 (D, D) is strictly related to the R-separarability, in the following we denote this game by R-separable(X).Similarly, the game G f in (D, D) is strictly related to the M-separability and in the following we denote this game Mseparable(X).This games were largely studied and some important characterizations of "Alice does not have a winning strategy"and "Bob has a winning strategy"have been given [1,15,17,18].Despite this some questions are still open.We denote by Bob ↑ G on X, the fact that "Bob has a winning strategy in the game G on X"and by Alice ↑ G on X, the fact that "Alice does not have a winning strategy in the game G on X".Remark 1.2.In general the following implications hold.
For some properties the last two implications of points 3 and 4 are, in fact, characterizations, that is Alice ↑ G(A, B) ⇐⇒ S(A, B).
In [12] it is proved that a space X is Rothberger if, and only if, Alice ↑ Rothberger(X).In [17,11] it is proved that if X is a space in which each point is a G δ , Bob ↑ Rothberger(X) if, and only if, X is countable.Similar arguments are valid for the Menger case: in [14,16] it is proved that a space X is Menger if, and only if, Alice ↑ Menger(X) and in [18] that if X is a metrizable space, Bob ↑ Menger(X) if, and only if, X is σ-compact.In [15] it is proved that Bob↑ R-separable(X) if, and only if, πw(X) = ω and, under CH, it is given an example of a R-separable space X such that Alice ↑ R-separable(X).
Two topological games G and G ′ are called dual if both "Alice ↑ G ⇐⇒ Bob ↑ G ′ "and "Alice ↑ G ′ ⇐⇒ Bob ↑ G"hold.Sometimes this dual vision could be useful to apply different techniques in demonstrations.For instance, the Point-open game is the dual of the Rothberger game (see [11]), the Point-picking game is the dual of G 1 (D, D) (see [15]), the Compact-open game is a possible dual of the Menger game (see [18]), but the question about the hypotesis to add to let them be dual is still open.
In Section 2 we study the R-nw-selective game.We present a characterization of the "Bob having a winning strategy"property and a sufficient condition for "Alice not having a winning"strategy property.We also give a consistent characterization, in terms of games, of the R-nw-selective property in the class of spaces without isolated points, with countable netweight and weight strictly less than cov(M).Moreover, we introduce the (Point, Open)-Set game and we prove that it is a promising candidate to be the dual of the R-nw-selective game.
In Section 3 we study the M-nw-selective game.We present, under some consistent hypotesis, a sufficient condition for "Alice not having a winning strategy"property and some necessary conditions for the "Bob having a winning strategy"property.We also give a consistent characterization, in terms of games, of the M-nw-selective property in the class of spaces with countable netweight and weight strictly less than d.
2 The R-nw-selective game Definition 2.1.Let X be a space with nw(X) = ω.The R-nw-selective game, denoted by R-nw-selective(X), is played according to the following rules.Alice chooses a countable network N 0 and Bob answers picking an element N 0 ∈ N 0 .Then Alice chooses another countable network N 1 and Bob answers in the same way and so on for countably many innings.At the end Bob wins if the set {N n : n ∈ ω} of his selections is a network.
Simultaneously we consider the possible dual version of the R-nw-selective game.
Definition 2.2.The (Point, Open)-Set game on a space X, denoted by POset(X), is played according to the following rules.Alice chooses a point x 0 and an open set U 0 containing x 0 .Then Bob picks N 0 a subset of X such that x 0 ∈ N 0 ⊆ U 0 .The game go ahead in this way for every n ∈ ω and Alice wins if the set {N n : n ∈ ω} of Bob's choices is a network.Proposition 2.3.Let X be a space.Bob ↑ R-nw-selective(X) if, and only if, the space X is countable and second countable.
Proof.Let M be the collection of all countable networks of X.Let σ a winning strategy for Bob.First we prove that the space is countable.Claim 1.
Indeed, suppose to have two points, say x and y, that are in all the clousure of the possible answers to N , for any N ∈ M. Since the space is T 1 , there exist a closed set C x containing x and not y, and a closed set C y containing y and not x.Pick a countable network then it is possible to construct a countable network such that any elements of it contains at most one of the points.Claim 2. There exists a countable , the complements of all the closures form an open cover of X \{x} (or X) and then, since having countable network implies hereditary Lindelöfness, we can obtain a countable subcover of X \ {x} (or of the all space X).Note that it is straightforward to prove the claims 1. and 2. for any inning.Consider the following tree of possible evolution of the R-nw-selective game on X.By claim 2. there exists (N n ∅ ) n∈ω , that is countably many possible choices of Alice in the first inning N 0 ∅ , N 1 ∅ , N 2 ∅ , ..., such that N ∈M σ(N ) = n∈ω σ(N n ∅ ).Fix, for example, the branch with N 0 ∅ then there exists a sequence ( ) is not empty we call this element x <0,1> , and so on.We obtain a subset X 0 = {x s : s ∈ ω <ω } and now we want to prove that X 0 = X.By contradiction, assume there exists y ∈ X \ X 0 .Then y / ∈ n∈ω σ(N n ∅ ); hence there exists an element of the sequence {σ(N n ∅ ) : n ∈ ω}, say σ(N k 0 ∅ ), such that y does not belong to it.By hypotesis, y / ∈ n∈ω σ(N k 0 ∅ , N n <k 0 > ); hence there exists an element of {σ(N n <k 0 > ) : n ∈ ω}, say σ(N k 1 <k 0 > ), such that y does not belong to it.Again, y ∈ n∈ω σ(N k 0 ∅ , N k 1 <k 0 > , N n <k 0 ,k 1 > ), there exists an element of {σ(N n <k 0 ,k 1 > ) : n ∈ ω}, say σ(N k 2 <k 0 ,k 1 > ), such that y does not belong to it.Proceeding in this way we obtain a branch consisting of elements that do not contain y; a contradiction, because such a branch is a network because σ is a winning strategy for Bob.Then X is countable.Now we prove that X is secound countable.Claim 3. If N ∈M σ(N ) = {x}, there exists an open set V such that x ∈ V ⊂ N ∈M σ(N ).Indeed, assume by contradiction that for every open set V such that x ∈ V there exists y V ∈ V \ σ(N ), for every N ∈ M. Let N be a countable network and consider the family Recall that, by Claim 2 there exists a countable subset M * ⊂ M such that N ∈M σ(N ) = N ∈M * σ(N ); further, since X is countable, N ∈M σ(N ) is countable and then there exists a countable subset M * * ⊂ M such that Even Claims 3 and 4 are clearly true for each inning.Consider the construction of the tree in the previous part of the proof.We know that as in Claim 3 and so on.Now we prove that {V s : s ∈ ω <ω } is a base.Assuming the contrary, there exist x ∈ X and an open set A with x ∈ A such that for every s ∈ ω <ω such that x ∈ V s , V s is not contained in A.
In the first inning, we have a family M ′ of countably many networks obtained as in Claim 4. Consider the intersection then we can pick, if there exists a N ∈ M ′ , such that x ∈ σ(N ), otherwise by hypothesis and Claim 3 we can pick a N ∈ M ′ , such that σ(N ) is not contained in A. Then, proceeding in this way for each inning, we find a branch of the tree, i.e., a R-nw-selective(X) in which Alice has a winning strategy, a contradiction.
The following proposition shows that the (Point, Open)-set game is a good candidate to be the dual of the R-nw-selective game.
Proposition 2.4.Let X be a space.The following implications hold.
Proof.The proof of points 1. and 2. is trivial.By Proposition 2.3 it is straightforward to prove the point 3.
Now we study the determinacy of the R-nw-selective game.
Proof.Suppose, by contradiction, that σ is a winning strategy for Alice in the R-nw-selective(X) and fix a base B of cardinality w(X).Construct a countable tree using the strategy σ in such a way that σ( ) = N 0 ; for each N 0 ∈ N 0 apply the strategy and so on.Look at this tree as the poset of all finite branches ordered with the inverse natural extention.The nodes in this tree are the countable networks that are images through the function σ.Fix x ∈ X and B ∈ B containing x, the set D (x,B) of all the finite sequences of the tree such that there exists an element of the sequence that is a σ( ..., N ) with x ∈ N ⊂ B, is dense in the poset.Since the cardinality of the family {D (x,B) : x ∈ X and B ∈ B} is less than cov(M) there exists a generic filter whose union is a branch of the tree and that intersects all the dense sets of the family.Therefore we obtain the contradiction, in particular this branch is a R-nw-selective(X) in which Bob wins.
Question 2.8.Is there any ZFC example of a space in which the R-nwselective game turns out to be indeterminate?
The following diagram shows all the relations found above.
Question 2.9.Does R-nw-selective property on a space X imply Alice ↑ R-nw-selective(X)?
Recall the following result.
Proposition 2.10.[8] Let X be a space without isolated points, with nw(X) = ω and w(X) < cov(M).Then the following are equivalent.
Then it is possible to give a partial answer to Question 2.9.
Proposition 2.11.Let X be a space without isolated points, with nw(X) = ω and w(X) < cov(M).Then the following are equivalent.
Now we show that if Bob is forced to select a fixed number of element from each network the obtained game is equivalent to the R-nw-selective game.Let Nw denote the class of all countable networks of a fixed space X.Let k ∈ ω and G k (Nw, Nw) on X be the game played in the following way: Alice chooses a countable network N 0 and Bob answers picking a subset F 0 ⊂ N 0 such that |F 0 | = k.Then Alice chooses another countable network N 1 and Bob answers picking a subset F 1 ⊂ N 1 such that |F 1 | = k and so on for countably many innings.At the end Bob wins if the set {F n : n ∈ ω} of his selections is a network.Proposition 2.12.Alice ↑ R-nw-selective(X) if, and only if, Alice ↑ G k (Nw, Nw) on X.
Proof.We only prove that if Alice ↑ R-nw-selective(X) then Alice ↑ G k (Nw, Nw) on X. Suppose that Alice selects a network N 0 in the G k (Nw, Nw).Then Bob answers by picking F 0 = {F 0 1 , ..., F 0 k }.Let Alice play the network N 0 in the R-nw-selective(X) for k-many innings and assume that Bob chooses F 0 i , i = 1, ..., k, in the i-th inning.Now suppose that Alice selects a network N 1 in the G k (Nw, Nw).Then Bob answers by picking F 1 = {F 1  1 , ..., F 1 k }.Let Alice play the network N 1 in the R-nw-selective(X) for k-many innings and assume that Bob chooses F 1 i , i = 1, ..., k, in the (k + i)-th inning and so on.Then by hypotesis n∈ω F n is not a network.Proposition 2.13.Bob ↑ R-nw-selective(X) if, and only if, Bob ↑ G k (Nw, Nw) on X.
Proof.We prove that Bob ↑ G k (Nw, Nw) on X implies that the space X is countable and secound countable.In fact the proof is similar to the one of Proposition 2.3.Let σ be a winning strategy for Bob in the G k (Nw, Nw) on X and M be the collection of all countable networks of the space X.We just need to prove the following claims.Claim 1. | N ∈M σ(N )| ≤ k.We prove the claim for k = 2. Suppose, by contradiction, that there are three different points x 1 , x 2 , x 3 ∈ N ∈M σ(N ).Since the space is T 1 , there exist a closed set C 1 containing x 1 but not x 2 , x 3 , a closed set C 2 containing x 2 but not x 1 , x 3 and a closed set C 3 containing x 3 but not x 1 , x 2 .Pick a countable network N .For every N ∈ N such that x 1 , x 2 , x 3 ∈ N, replace N with N ∩ C i , i = 1, 2, 3.In this way we construct a countable network such 2. Alice ↑ M-nw-selective(X); 3. X is M-nw-selective.However, it is worthwhile to pose the following question.Question 3.5.Does the M-nw-selective property imply that Alice ↑ Mnw-selective(X)?Proposition 3.6.Let X be a regular space such that Bob ↑ M-nw-selective(X).Then X is σ-compact.
Proof.Let M be the collection of all countable networks of X and σ a winning strategy for Bob in M-nw-selective(X).
The proof is similar to the one of Claim 2. in Proposition 2.3 and, as in there, these claims are true also for all the other innings.There exists (N n ∅ ) n∈ω , that is countably many possible first inning ) is a compact subset, we call this element K <0,1> , and so on.Consider the set X 0 = {K s : s ∈ ω <ω }.Now we prove that X 0 = X.By contradiction, assume there exists y ∈ X \ X 0 .Then y / ∈ n∈ω σ(N n ∅ ); hence there exists n 0 ∈ ω such that y ∈ σ(N n 0 ∅ ).Again, y / ∈ n∈ω σ(N n 0 ∅ , N n n 0 ); hence there exists n 1 ∈ ω such that y ∈ σ(N n 0 ∅ , N n 1 n 0 ).Proceeding in this way we obtain a branch (or an evolution of the M-nw-selective(X)) in which Bob does not win, a contradiction, because σ is a winning strategy.Then X is σ-compact.
Corollary 3.7.Let X be a regular space in which Bob ↑ M-nw-selective(X).Then X is secound countable.
Recall that every regular secound countable space is metrizable and a space is called σ-(metrizable compact) if it is union of countably many metrizable compact spaces.Then it is possible to obtain the following corollary.
Corollary 3.8.Let X be a regular space in which Bob ↑ M-nw-selective(X).Then X is metrizable.Corollary 3.9.Let X be a regular space in which Bob ↑ M-nw-selective(X).Then X is σ-(metrizable compact).
The following is a consistent example showing that the M-nw-selective game can be indeterminate.It also shows that the vice versa of the Corollaries 3.7 and 3.8 does not hold.The hypotesis of regularity in Corollary 3.7 can be replaced by countability of the space as the following shows.Proposition 3.11.If X is a countable space in which Bob ↑ M-nw-selective(X).Then X is secound countable.
where τ x denotes the family of all open sets containing x andN x = {N ∈ N : x ∈ N}.Since X is countable, N ′ is countable.Now we prove that N ′ is a network.Let y ∈ X, y = x,and A be an open set such that y ∈ A. Since X is T 2 , there exists an open set B such that y ∈ B and x / ∈ B. Then there exists N ∈ N such that y ∈ N ⊂ A ∩ B. Therefore N ∈ N \ N x .Claim 4. If N ∈M σ(N ) = {x}, there exists M ′ ⊂ M countable such that N ∈M ′ σ(N ) = {x} and also such that N ∈M ′ σ(N ) = N ∈M σ(N ).
Claim 1: N ∈M σ(N ) is compact.Indeed, put K = N ∈M σ(N ), let U be a cover made by open sets of X and N ∈ M. Consider the network N ′ = {N ∈ N : N ⊂ U for some U ∈ U} ∪ {N ∈ N : N ∩ K = ∅}.Then K ⊂ σ( N ′ ) and considering the corresponding open sets we have the compactness of K. Claim 2: There exists a countable subset M
By Claim 1, each intersection is empty or contains only one element.If the intersection n∈ω σ(N n ∅ ) is non-empty we call this element x ∅ , otherwise we go on; if n∈ω σ(N 0 ∅ , N n <0> ) is not empty we call this element x <0> ; if the intersection n∈ω By claims 1 and 2, each intersection, if it is not empty, is a compact subset.If the intersection is empty, we do not do anything and if n∈ω