On open-open games of uncountable length

The aim of this note is to investigate the open-open game of uncountable length. We introduce a cardinal number $\mu(X)$, which says how long the Player I has to play to ensure a victory. It is proved that $\su(X)\leq\mu(X)\leq\su(X)^+$. We also introduce the class $\mathcal C_\kappa$ of topological spaces that can be represented as the inverse limit of $\kappa$-complete system $\{X_\sigma,\pi^\sigma_\rho,\Sigma\}$ with $\w(X_\sigma)\leq\kappa$ and skeletal bonding maps. It is shown that product of spaces which belong to $\mathcal C_\kappa$ also belongs to this class and $\mu(X)\leq\kappa$ whenever $X\in\mathcal C_\kappa$ .


Introduction
The following game is due to P. Daniels, K. Kunen and H. Zhou In this note, we consider what happens if one drops restrictions on the length of games. If κ is an infinite cardinal and rounds are played for every ordinal number less then κ, then this modification is called the open-open game of length κ. The examination of such games is a continuation of [9], [10] and [11]. A cardinal number µ(X) is introduced such that c(X) ≤ µ(X) ≤ c(X) + . Topological spaces, which can be represented as an inverse limit of κ-complete system {X σ , π σ ̺ , Σ} with w(X σ ) ≤ κ and each X σ is T 0 space and skeletal bonding map π σ ̺ , are listed as the class C κ . If µ(X) = ω, then X ∈ C ω . There exists a space X with X ∈ C µ(X) . The class C κ is closed under any Cartesian product. In particular, the cellularity number of X I is equal κ whenever X ∈ C κ . This implies Theorem of D. Kurepa that c(X I ) ≤ 2 κ , whenever c(X) ≤ κ. Undefined notions and symbols are used in accordance with books [3], [5] and [8]. For example, if κ is a cardinal number, then κ + denotes the first cardinal greater than κ.

When games favor Player I
Let X be a topological space. Denote by T the family of all nonempty open sets of X. For an ordinal number α, let T α denotes the set of all sequences of the length α consisting of elements of T . The space X is called κ-favorable whenever there exists a function s : {(T ) α : α < κ} → T such that for each sequence {B α+1 : α < κ} ⊆ T with B 1 ⊆ s(∅) and B α+1 ⊆ s({B γ+1 : γ < α}), for each α < κ, the union {B α+1 : α < κ} is dense in X. We may also say that the function s is witness to κfavorability of X. In fact, s is a winning strategy for Player I. For abbreviation we say that s is κ-winning strategy. Sometimes we do not precisely define a strategy. Just give hints how a player should play. Note that, any winning strategy can be arbitrary on steps for limit ordinals.
A family B of open non-empty subset is called a π-base for X if every non-empty open subset U ⊆ X contains a member of B. The smallest cardinal number |B|, where B is a π-base for X, is denoted by π(X). Proposition 1. Any topological space X is π(X)-favorable.
Proof. Let {U α : α < π(X)} be a π-base. Put s(f ) = U α for any sequence f ∈ T α . Each family {B γ : B γ ⊆ U γ and γ < π(X)} of open non-empty sets is again a π-base for X. So, its union is dense in X.
According to [5, p. 86] the cellularity of X is denoted by c(X). Let sat(X) be the smallest cardinal number κ such that every family of pairwise disjoint open sets of X has cardinality < κ, compare [6]. Clearly, if sat(X) is a limit cardinal, then sat(X) = c(X). In all other cases, sat(X) = c(X) + . Hence, c(X) ≤ sat(X) ≤ c(X) + . Let µ(X) = min{κ : X is a κ-favorable and κ is a cardinal number}.
There exists a separable space X which is not ω 0 -favorable, see A. Szymański [14] or [4, p.207-208]. Hence we get ω 0 = c(X) < µ(X) = sat(X) = ω 1 .  According to [3], a directed set Σ is said to be κ-complete if any chain of length ≤ κ consisting of its elements has the least upper bound in Σ. An inverse system {X σ , π σ ̺ , Σ} is said to be a κ-complete, whenever Σ is κ-complete and for every chain A ⊆ Σ, where |A| ≤ κ, such that σ = sup A ∈ Σ we get

On inverse systems with skeletal bonding maps
In addition, we assume that bonding maps are surjections.
For ω-favorability, the following lemma is given without proof in [4,Corollary 1.4]. We give a proof to convince the reader that additional assumptions on topology are unnecessary.
Proof. Let a function σ X be a witness to κ-favorability of X. Put The next Theorem is similar to [2, Theorem 2]. We replace a continuous inverse system with indexing set being a cardinal, by κ-complete inverse system, and also c(X) is replaced by µ(X). Let κ be a fixed cardinal number.
Theorem 5. Let X be a dense subset of the inverse limit of the κ- If all bonding maps are skeletal, then µ(X) = κ.
Proof. By Lemma 4, one can assume that X = lim ← − {X σ , π σ ̺ , Σ}. Fix functions s σ : T <κ σ → T σ , each one is a witness to µ(X σ )-favorability of X σ . This does not reduce the generality, because µ(X σ ) ≤ κ for every σ ∈ Σ. In order to explain the induction, fix a bijection f : κ → κ × κ such that: One can take as f an isomorphism between κ and κ × κ, with canonical well-ordering, see [8]. The function f will indicate the strategy and sets that we have taken in the following induction.
We construct a function s : T <κ → T which will provide κfavorability of X. The first step is defined for f (0) = (0, 0). Take an arbitrary σ 1 ∈ Σ and put
The above corollary is similar to [2, Theorem 1], but we replaced a continuous inverse system, whose indexing set is a cardinal number by κ-complete inverse system.

Classes C κ
Let κ be an infinite cardinal number. Consider inverse limits of κcomplete system {X σ , π σ ̺ , Σ} with w(X σ ) ≤ κ. Let C κ be a class of such inverse limits with skeletal bonding maps and X σ being T 0 -space. Now, we show that the class C κ is stable under Cartesian products.
Theorem 7. The Cartesian product of spaces from C κ belongs to C κ .
Proof. Let X = {X s : s ∈ S} where each X s ∈ C κ . For each s ∈ S, let X s = lim ← − {X σ , s σ ρ , Σ s } be a κ-complete inverse system with skeletal bonding map such that each T 0 -space X σ has the weight ≤ κ. Consider the union Introduce a partial order on Γ as follows: where ≤ a is the partial order on Σ a . The set Γ with the relation is upward directed and κ-complete.
If f ∈ Γ, then Y f denotes the Cartesian product If f g, then put f (a) is the Cartesian product of the bonding maps a g(a) f (a) : X g(a) → X f (a) . We get the inverse system{Y f , p g f , Γ} which is κ-complete, bonding maps are skeletal and w(Y f ) ≤ κ. So, we can take Y = lim ← − {Y f , p g f , Γ}. Now, define a map h : X → Y by the formula: the map h is well defined and it is injection.
. We shall prove that an element b t is a thread of the space X t . Indeed, if σ ≥ ρ and σ, ρ ∈ Σ t , then take functions f t σ and g t ρ .
For abbreviation, denote f = f t σ and g = g t ρ . Define a function h : dom(f ) ∪ dom(g) → {Σ t : t ∈ dom(f ) ∪ dom(g)} in the following way: The function h is element of Γ and f, g h. Note that h| dom(f ) = f and h| dom( We shall prove that the map h is continuous. Take an open subset is open subset. A map p f is projection from the inverse limit Y to Y f . It is sufficient to show that : Since the map h is bijection and for any subbase subset s∈S B s ⊆ X, the map h is open. In the case κ = ω we have well know results that I-favorable space is productable (see [4] or [9])

Corollary 8. Every I-favorable space is stable under any product.
If D is a set and κ is cardinal number then we denote α<κ D α by D <κ .
The following result probably is known but we give a proof for the sake of completnes.
Theorem 9. Let κ be an infinite cardinal and let T be a set such that |T | ≥ κ κ . If A ∈ [T ] κ and f δ : T <κ → T for all δ < κ <κ then there exists a set B ⊆ T such that |B| ≤ τ and A ⊆ B and f δ (C) ∈ B for every C ∈ B <κ and every δ < κ <κ , where τ = κ <κ for regular κ κ κ otherwise.
In the second case cf(κ) < κ, we proceed the above induction up to β = κ. Let B = A κ , so we get |B| ≤ κ κ and B = β<κ + A β . Similarly to the first case we get that B is closed under all function f δ , δ < κ <κ .
Theorem 10. If X belongs to the class C κ then c(X) ≤ κ .
We apply some facts from the paper [10]. Let P be a family of open subset of topological space X and x, y ∈ X. We say that x ∼ P y if and only if x ∈ V ⇔ y ∈ V for every V ∈ P. The family of all sets [x] P = {y : y ∼ P x} we denote by X/P. Define a map q : X → X/P as follows q[x] = [x] P . The set X/P is equipped with topology T P generated by all images q[V ] where V ∈ P.
Recall Lemma 1 from paper [10]: If P is a family of open set of X and P is closed under finite intersection then the mapping q : X → X/P is continuous. Moreover if X = P then the family {q[V ] : V ∈ P} is a base for the topology T P .
Theorem 11. If P is a set of open subset of topological space X such that: (1) is closed under κ-winning strategy, finite union and finite intersection, (2) has property (seq), then X/P with topology T P is completely regular space and q : X → X/P is skeletal.
If a topological space X has the cardinal number µ(X) = ω then X ∈ C ω , but for µ(X) equals for instance ω 1 we do not even know if X ∈ C ω 1 ω .
Proof. Let B be a π-base for topological space X consisting of cozerosets and σ : {B α : α < κ} → B be a κ-winning strategy. We can define a function of finite intersection property and finite union property as following : . By Theorem 9 for each R ∈ [B] κ and all functions h, g, σ n , σ there is subset P ⊆ B such that: (1) |P| ≤ τ where τ = κ <κ for regular κ κ κ otherwise, (2) R ⊆ P, (3) P is closed under κ-winning strategy σ, function of finite intersection property and finite union property , (4) P is closed under σ n , n < ω, hence P holds property (seq).
Therefore by Theorem 11 we get skeletal mapping q P : X → X/P. Let Σ ⊆ [B] ≤τ be a set of families which satisfies above condition (1), (2), (3), (4). If Σ is directed by inclusion. It is easy to check that Σ is τ -complete. Similar to [10,Theorem 11] we define a function f : X → Y as following f (x) = {f P (x)}, where f (x) P = q P (x) and Y = lim ← − {X/R, q R P , C}. If R, P ∈ C and P ⊆ R, then q R P (f (x) R ) = f (x) P . Thus f (x) is a thread, i.e. f (x) ∈ Y . It easy to see that f is homeomorphism onto its image and f [X] is dense in Y , compare [10, proof of Theorem 11] The Theorem 12 suggests a question: Does each space X belong to C µ(X) ?
Proof. By Theorem 9 we get X I ∈ C τ . Hence by Theorem 10 and 7 we have c(X I ) ≤ τ .
By above Corollary we get the following