On Open-Open Games of Uncountable Length

Abstract

The aim of this paper is to investigate the open-open game of uncountable length. We introduce a cardinal number 𝜇(𝑋), which says how long the Player I has to play to ensure a victory. It is proved that 𝑐(𝑋)≤𝜇(𝑋)≤𝑐(𝑋)⁺. We also introduce the class 𝒞_𝜅 of topological spaces that can be represented as the inverse limit of 𝜅-complete system {𝑋_𝜎,𝜋^𝜎_𝜌,Σ} with 𝑤(𝑋_𝜎)≤𝜅 and skeletal bonding maps. It is shown that product of spaces which belong to 𝒞_𝜅 also belongs to this class and 𝜇(𝑋)≤𝜅 whenever 𝑋∈𝒞_𝜅.

1. Introduction

The following game is due to Daniels et al. [1]: two players take turns playing on a topological space 𝑋; a round consists of Player I choosing a nonempty open set 𝑈⊆𝑋 and Player II choosing a nonempty open set 𝑉⊆𝑈; a round is played for each natural number. Player I wins the game if the union of open sets which have been chosen by Player II is dense in 𝑋. This game is called the open-open game.

In this paper, 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 than 𝜅, then this modification is called the open-open game of length 𝜅. The examination of such games is a continuation of [2–4]. 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_𝜚,Σ} with 𝑤(𝑋_𝜎)≤𝜅 and each 𝑋_𝜎 is 𝑇₀ space and skeletal bonding map 𝜋^𝜎_𝜚, 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 Kurepa that 𝑐(𝑋^𝐼)≤2^𝜅, whenever 𝑐(𝑋)≤𝜅. Undefined notions and symbols are used in accordance with books [5–7]. For example, if 𝜅 is a cardinal number, then 𝜅⁺ denotes the first cardinal greater than 𝜅.

2. 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 denote 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, (2.1) such that for each sequence {𝐵_𝛼+1∶𝛼<𝜅}⊆𝒯 with 𝐵₁⊆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 𝜅-favorability 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 2.1. 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_𝛾and𝛾<𝜋(𝑋)} of open non-empty sets is again a 𝜋-base for 𝑋. So, its union is dense in 𝑋.

According to [6, 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 [8]. 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∶𝑋isa𝜅-favorableand𝜅isacardinalnumber}. (2.2) Proposition 2.1 implies 𝜇(𝑋)≤𝜋(𝑋). The next proposition gives two natural strategies and gives more accurate estimation than 𝑐(𝑋)≤𝜇(𝑋)≤𝑐⁺(𝑋).

Proposition 2.2. 𝑐(𝑋)≤𝜇(𝑋)≤sat(𝑋).

Proof. Suppose 𝑐(𝑋)>𝜇(𝑋). Fix a family {𝑈_𝜉∶𝜉<𝜇(𝑋)⁺} of pairwise disjoint open sets. If Player II always chooses an open set, which meets at most one 𝑈_𝜉, then he will not lose the open-open game of the length 𝜇(𝑋), a contradiction.
Suppose sets {𝐵_𝛾+1∶𝛾<𝛼} are chosen by Player II. If the set 𝑋⧵cl𝐵_𝛾+1∶𝛾<𝛼 (2.3) is non-empty, then Player I choses it. Player I wins the open-open game of the length sat(𝑋), when he will use this rule. This gives 𝜇(𝑋)≤sat(𝑋).

Note that, 𝜔₀=𝑐({0,1}^𝜅)=𝜇({0,1}^𝜅)≤sat({0,1}^𝜅)=𝜔₁, where {0,1}^𝜅 is the Cantor cube of weight 𝜅. There exists a separable space 𝑋 which is not 𝜔₀-favorable, see Szymański [9] or [1, p.207-208]. Hence we get 𝜔₀=𝑐(𝑋)<𝜇(𝑋)=sat(𝑋)=𝜔₁. (2.4)

3. On Inverse Systems with Skeletal Bonding Maps

Recall that, a continuous surjection is skeletal if for any non-empty open sets 𝑈⊆𝑋 the closure of 𝑓[𝑈] has non-empty interior. If 𝑋 is a compact space and 𝑌 is a Hausdorff space, then a continuous surjection 𝑓∶𝑋→𝑌 is skeletal if and only if Int𝑓[𝑈]≠∅, for every non-empty and open 𝑈⊆𝑋, see Mioduszewski and Rudolf [10].

Lemma 3.1. A skeletal image of 𝜅-favorable space is a 𝜅-favorable space.

Proof. A proof follows by the same method as in [11, Theorem 4.1]. In fact, repeat and generalize the proof given in [4, Lemma 1].

According to [5], 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_𝜚,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=sup𝐴∈Σ we get 𝑋_𝜎=lim_←𝑋_𝛼,𝜋^𝛽_𝛼,𝐴. (3.1) In addition, we assume that bonding maps are surjections.

For 𝜔-favorability, the following lemma is given without proof in [1, Corollary 1.4]. We give a proof to convince the reader that additional assumptions on topology are unnecessary.

Lemma 3.2. If 𝑌⊆𝑋 is dense, then 𝑋 is 𝜅-favorable if and only if 𝑌 is 𝜅-favorable.

Proof. Let a function 𝜎_𝑋 be a witness to 𝜅-favorability of 𝑋. Put 𝜎_𝑌(∅)=𝜎_𝑋(∅)∩𝑌. (3.2) If Player II chooses open set 𝑉₁∩𝑌⊆𝜎_𝑌(∅), then put 𝑉^′₁=𝑉₁∩𝜎_𝑋(∅)⊆𝜎_𝑋(∅). (3.3) We get 𝑉^′₁∩𝑌=𝑉₁∩𝑌⊆𝜎_𝑌(∅), since 𝑉₁∩𝑌⊂𝜎_𝑋(∅)∩𝑌. Then we put 𝜎_𝑌𝑉₁∩𝑌=𝜎_𝑋𝑉^′₁∩𝑌. (3.4) Suppose we have already defined 𝜎_𝑌𝑉_𝛼+1∩𝑌∶𝛼<𝛾=𝜎_𝑋𝑉^′_𝛼+1∶𝛼<𝛾∩𝑌, (3.5) for 𝛾<𝛽<𝜅. If Player II chooses open set 𝑉_𝛽+1∩𝑌⊆𝜎_𝑌({𝑉_𝛼+1∩𝑌∶𝛼<𝛽}), then put 𝑉^′_𝛽+1=𝑉_𝛽+1∩𝜎_𝑋𝑉^′_𝛼+1∶𝛼<𝛽⊆𝜎_𝑋𝑉^′_𝛼+1∶𝛼<𝛽. (3.6) Finally, put 𝜎_𝑌𝑉_𝛼+1∩𝑌∶𝛼≤𝛽=𝜎_𝑋𝑉^′_𝛼+1∶𝛼≤𝛽∩𝑌 (3.7) and check that 𝜎_𝑌 is witness to 𝜅-favorability of 𝑌.
Assume that 𝜎_𝑌 is a witness to 𝜅-favorability of 𝑌. If 𝜎_𝑌(∅)=𝑈₀∩𝑌 and 𝑈₀⊆𝑋 is open, then put 𝜎_𝑋(∅)=𝑈₀. If Player II chooses open set 𝑉₁⊆𝜎_𝑋(∅), then 𝑉₁∩𝑌⊆𝜎_𝑌(∅). Put 𝜎_𝑋(𝑉₁)=𝑈₁, where 𝜎_𝑌(𝑉₁∩𝑌)=𝑈₁∩𝑌 and 𝑈₁⊆𝑋 is open. Suppose 𝜎_𝑌𝑉_𝛼+1∩𝑌∶𝛼<𝛾=𝑈_𝛾∩𝑌,𝜎_𝑋𝑉_𝛼+1∶𝛼<𝛾=𝑈_𝛾 (3.8) have been already defined for 𝛾<𝛽<𝜅. If II Player chooses open set 𝑉_𝛽+1⊆𝜎_𝑋({𝑉_𝛼+1∶𝛼<𝛽}), then put 𝜎_𝑋({𝑉_𝛼+1∶𝛼<𝛽+1})=𝑈_𝛽+1, where open set 𝑈_𝛽+1⊆𝑋𝑋 is determined by 𝜎_𝑌({𝑉_𝛼+1∩𝑌∶𝛼<𝛽+1})=𝑈_𝛽+1∩𝑌.

The next theorem is similar to [12, 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 3.3. Let 𝑋 be a dense subset of the inverse limit of the 𝜅-complete system {𝑋_𝜎,𝜋^𝜎_𝜚,Σ}, where 𝜅=sup{𝜇(𝑋_𝜎)∶𝜎∈Σ}. If all bonding maps are skeletal, then 𝜇(𝑋)=𝜅.

Proof. By Lemma 3.2, one can assume that 𝑋=lim_←{𝑋_𝜎,𝜋^𝜎_𝜚,Σ}. Fix functions 𝐬_𝜎∶𝒯_𝜎^<𝜅→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 (1)if 𝑓(𝛼)=(𝛽,𝜁), then 𝛽,𝜁≤𝛼; (2)𝑓⁻¹(𝛽,𝛾)<𝑓⁻¹(𝛽,𝜁) if and only if 𝛾<𝜁; (3)𝑓⁻¹(𝛾,𝛽)<𝑓⁻¹(𝜁,𝛽) if and only if 𝛾<𝜁.
One can take as 𝑓 an isomorphism between 𝜅 and 𝜅×𝜅, with canonical well-ordering, see [7]. 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 𝜅-favorability of 𝑋. The first step is defined for 𝑓(0)=(0,0). Take an arbitrary 𝜎₁∈Σ and put 𝐬(∅)=𝜋_𝜎⁻¹₁𝐬_𝜎1(∅). (3.9) Assume that Player II chooses non-empty open set 𝐵₁=𝜋_𝜎⁻¹₂(𝑉₁)⊆𝐬(∅), where 𝑉₁⊆𝑋_𝜎2 is open. Let 𝐬𝐵₁=𝜋_𝜎⁻¹₁𝐬_𝜎1Intcl𝜋_𝜎1𝐵₁∩𝐬_𝜎1(∅) (3.10) and denote 𝐷⁰₀=Intcl𝜋_𝜎1(𝐵₁)∩𝐬_𝜎1(∅). So, after the first round and the next respond of Player I, we know: indexes 𝜎₁ and 𝜎₂, the open set 𝐵₁⊆𝑋 and the open set 𝐷⁰₀⊆𝑋_𝜎1.
Suppose that sequences of open sets {𝐵_𝛼+1⊆𝑋∶𝛼<𝛾}, indexes {𝜎_𝛼+1∶𝛼<𝛾}, and sets {𝐷^𝜑_𝜁∶𝑓⁻¹(𝜑,𝜁)<𝛾} have been already defined such that.
If 𝛼<𝛾 and 𝑓(𝛼)=(𝜑,𝜂), then 𝐵_𝛼+1=𝜋_𝜎⁻¹_𝛼+2𝑉_𝛼+1⊆𝐬𝐵_𝜉+1∶𝜉<𝛼=𝜋_𝜎⁻¹_𝜑+1𝐬_𝜎𝜑+1𝐷^𝜑_𝜈∶𝜈<𝜂, (3.11) where 𝐷^𝜑_𝜈=Intcl𝜋_𝜎𝜑+1(𝐵_𝑓⁻¹_{((𝜑,𝜈))+1})∩𝐬_𝜎𝜑+1({𝐷^𝜑_𝜁∶𝜁<𝜈}) and 𝑉_𝛼+1⊆𝑋_𝜎𝛼+2 are open.
If 𝑓(𝛾)=(𝜃,𝜆) and 𝛽<𝜆, then take 𝐷^𝜃_𝛽=Intcl𝜋_𝜎𝜃+1𝐵_𝑓⁻¹_{((𝜃,𝛽))+1}∩𝐬_𝜎𝜃+1𝐷^𝜃_𝜁∶𝜁<𝛽 (3.12) and put 𝐬𝐵_𝛼+1∶𝛼<𝛾=𝜋_𝜎⁻¹_𝜃+1𝐬_𝜎𝜃+1𝐷^𝜃_𝛼∶𝛼<𝜆. (3.13) Since Σ is 𝜅-complete, one can assume that the sequence {𝜎_𝛼+1∶𝛼<𝜅} is increasing and 𝜎=sup{𝜎_𝜉+1∶𝜉<𝜅}∈Σ.
We will prove that ⋃_𝛼<𝜅𝐵_𝛼+1 is dense in 𝑋. Since 𝜋_𝜎⁻¹(𝜋_𝜎(𝐵_𝛼+1))=𝐵_𝛼+1 for each 𝛼<𝜅 and 𝜋_𝜎 is skeletal map, it is sufficient to show that ⋃_𝛼<𝜅𝜋_𝜎(𝐵_𝛼+1) is dense in 𝑋_𝜎. Fix arbitrary open set (𝜋^𝜎_𝜎𝜉+1)⁻¹(𝑊) where 𝑊 is an open set of 𝑋_𝜉+1. Since 𝐬_𝜎𝜉+1 is winning strategy on 𝑋_𝜎𝜉+1, there exists 𝐷^𝜉_𝛼 such that 𝐷^𝜉_𝛼∩𝑊≠∅, and 𝐷^𝜉_𝛼⊆Intcl𝜋_𝜎𝜉+1(𝐵_𝑓⁻¹_{((𝜉,𝛼))+1}). Therefore we get 𝜋^𝜎_𝜎𝜉+1⁻¹(𝑊)∩𝜋_𝜎𝐵_𝛿+1≠∅, (3.14) where 𝛿=𝑓⁻¹((𝜉,𝛼)). Indeed, suppose that (𝜋^𝜎_𝜎𝜉+1)⁻¹(𝑊)∩𝜋_𝜎(𝐵_𝛿+1)=∅. Then ∅=𝜋^𝜎_𝜎𝜉+1𝜋^𝜎_𝜎𝜉+1⁻¹(𝑊)∩𝜋_𝜎𝐵_𝛿+1=𝑊∩𝜋^𝜎_𝜎𝜉+1𝜋_𝜎𝐵_𝛿+1=𝑊∩𝜋_𝜎𝜉+1𝐵_𝛿+1. (3.15)
Hence we have 𝑊∩Intcl𝜋_𝜎𝜉+1(𝐵_𝛿+1)=∅, a contradiction.

Corollary 3.4. If 𝑋 is dense subset of an inverse limit of 𝜇(𝑋)-complete system {𝑋_𝜎,𝜋^𝜎_𝜚,Σ}, where all bonding map are skeletal, then 𝑐(𝑋)=sup𝑐𝑋_𝜎∶𝜎∈Σ. (3.16)

Proof. Let 𝑋=lim_←{𝑋_𝜎,𝜋^𝜎_𝜚,Σ}. Since 𝑐(𝑋)≥𝑐(𝑋_𝜎), for every 𝜎∈Σ, we will show that 𝑐(𝑋)≤sup𝑐𝑋_𝜎∶𝜎∈Σ. (3.17)
Suppose that sup{𝑐(𝑋_𝜎)∶𝜎∈Σ}=𝜏<𝑐(𝑋). Using Proposition 2.2 and Theorem 3.3, check that 𝜇(𝑋)=sup𝜇𝑋_𝜎∶𝜎∈Σ≤sup𝑐𝑋_𝜎⁺∶𝜎∈Σ≤𝜏⁺≤𝑐(𝑋). (3.18) So, we get 𝜇(𝑋)=𝑐(𝑋)=𝜏⁺. Therefore, there exists a family ℛ, of size 𝜏⁺, which consists of pairwise disjoint open subset of 𝑋. We can assume that ℛ⊆𝜋_𝜎⁻¹(𝑈)∶𝑈isanopensubsetof𝑋_𝜎,𝜎∈Σ. (3.19) Since {𝑋_𝜎,𝜋^𝜎_𝜚,Σ} is 𝜇(𝑋)-complete inverse system and |ℛ|=𝜇(𝑋), there exists 𝛽∈Σ such that ℛ⊆𝜋_𝛽⁻¹(𝑈)∶𝑈isanopensubsetof𝑋_𝛽, (3.20) a contradiction with 𝑐(𝑋_𝛽)<𝜏⁺.

The above corollary is similar to [12, Theorem 1], but we replaced a continuous inverse system, whose indexing set is a cardinal number by 𝜅-complete inverse system.

4. Classes 𝒞_𝜅

Let 𝜅 be an infinite cardinal number. Consider inverse limits of 𝜅-complete system {𝑋_𝜎,𝜋^𝜎_𝜚,Σ} with 𝑤(𝑋_𝜎)≤𝜅. Let 𝒞_𝜅 be a class of such inverse limits with skeletal bonding maps and 𝑋_𝜎 being 𝑇₀-space. Now, we show that the class 𝒞_𝜅 is stable under Cartesian products.

Theorem 4.1. 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 𝑇₀-space 𝑋_𝜎 has the weight ≤𝜅. Consider the union Γ=_𝑠∈𝐴Σ_𝑠∶𝐴∈[𝑆]^𝜅. (4.1) Introduce a partial order on Γ as follows: 𝑓⪯𝑔⟺dom𝑓⊆dom𝑔,∀_{𝑎∈dom(𝑓)}𝑓(𝑎)≤_𝑎𝑔(𝑎), (4.2) 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 𝑋_𝑓(𝑎)∶𝑎∈dom𝑓. (4.3) If 𝑓⪯𝑔, then put 𝑝^𝑔_𝑓=_{𝑎∈dom(𝑓)}𝑎^𝑔(𝑎)_𝑓(𝑎)∘𝜋_𝑓^dom(𝑔)_dom, (4.4) where 𝜋^dom(𝑔)_dom(𝑓) is the projection of ∏{𝑋_𝑔(𝑎)∶𝑎∈dom(𝑔)} onto ∏{𝑋_𝑔(𝑎)∶𝑎∈dom(𝑓)} and ∏_{𝑎∈dom(𝑓)}𝑎^𝑔(𝑎)_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 following formula: ℎ𝑥_{𝑠𝑠∈𝑆}=𝑥_{𝑓𝑓∈Γ}, (4.5) where 𝑥_𝑓={𝑥_𝑓(𝑎)}_{𝑎∈dom(𝑓)}∈𝑌_𝑓 and 𝑓∈∏{Σ_𝑎∶𝑎∈dom(𝑓)} and dom(𝑓)∈[𝑆]^𝜅. By the property 𝑥_{𝑠𝑠∈𝑆}=𝑡_{𝑠𝑠∈𝑆}⟺∀_𝑠∈𝑆∀_{𝜎∈Σ𝑠},𝑥_𝜎=𝑡_𝜎⟺∀_𝑓∈Γ,𝑥_𝑓=𝑡_𝑓, (4.6) the map ℎ is well defined and it is injection.
The map ℎ is surjection. Indeed, let {𝑏_𝑓}_𝑓∈Γ∈𝑌. For each 𝑠∈𝑆 and each 𝜎∈Σ_𝑠 we fix 𝑓^𝑠_𝜎∈Γ such that 𝑠∈dom(𝑓^𝑠_𝜎) and 𝑓^𝑠_𝜎(𝑠)=𝜎. Let 𝜋_𝑓(𝑠)∶𝑌_𝑓→𝑋_𝑓(𝑠) be a projection for each 𝑓∈Γ.
For each 𝑡∈𝑆 let define 𝑏_𝑡={𝑏_𝜎}_{𝜎∈Σ𝑡}, where 𝑏_𝜎=𝜋_𝑓^𝑡_𝜎(𝑡)(𝑏_𝑓^𝑡_𝜎). We will 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: ℎ(𝑠)=𝑔(𝑠)𝑔𝑓,𝑓(𝑠)𝑓.,if𝑠∈dom⧵dom,if𝑠∈dom (4.7) The function ℎ is element of Γ and 𝑓,𝑔⪯ℎ. Note that ℎ∣dom(𝑓)=𝑓 and ℎ∣dom(𝑔)⧵{𝑡}=𝑔∣dom(𝑔)⧵{𝑡}. Since 𝑏_{𝑔(𝑠)𝑠∈dom(𝑔)}=𝑏_𝑔=𝑝^ℎ_𝑔𝑏_ℎ=_{𝑔𝑠∈dom}𝑠_𝑔(𝑠)^ℎ(𝑠)𝜋_𝑔^dom(ℎ)_dom𝑏_ℎ=_{𝑔𝑠∈dom}𝑠_𝑔(𝑠)^ℎ(𝑠)𝑏_{ℎ(𝑠)𝑠∈dom(𝑔)}=𝑠_𝑔(𝑠)^ℎ(𝑠)𝑏_{ℎ(𝑠)𝑠∈dom(𝑔)}, (4.8) we get 𝑏_𝜌=𝑏_𝑔(𝑡)=𝑠_𝑔(𝑡)^ℎ(𝑡)𝑏_ℎ(𝑡)=𝑠_𝑔(𝑡)^𝑓(𝑡)𝑏_𝑓(𝑡)=𝑠^𝜎_𝜌𝑏_𝜎. (4.9) It is clear that ℎ({𝑎_𝑡}_𝑡∈𝑆)={𝑏_𝑓}_𝑓∈Γ.
We shall prove that the map ℎ is continuous. Take an open subset 𝑈=∏_{𝑠∈dom(𝑓)}𝐴_𝑓(𝑠)⊆𝑌_𝑓 such that 𝐴_𝑓(𝑠)=𝑉,if𝑠=𝑠₀,𝑋_𝑓(𝑠),otherwise, (4.10) where 𝑉⊆𝑋_𝑓(𝑠0) is open subset. A map 𝑝_𝑓 is projection from the inverse limit 𝑌 to 𝑌_𝑓. It is sufficient to show that ℎ⁻¹𝑝_𝑓⁻¹(𝑈)=_𝑠∈𝑆𝐵_𝑠, (4.11) where 𝐵_𝑠=𝑊,if𝑠=𝑠₀,𝑋_𝑠,otherwise, (4.12) and 𝑊=𝜋⁻¹_𝑓(𝑠0)(𝑉) and 𝜋_𝑓(𝑠0)∶𝑌_𝑓→𝑋_𝜎0 is the projection and 𝑓(𝑠₀)=𝜎₀. We have 𝑥_{𝑠𝑠∈𝑆}∈ℎ⁻¹𝑝_𝑓⁻¹(𝑈)⟺𝑝_𝑓ℎ𝑥_{𝑠𝑠∈𝑆}∈𝑈⟺𝑝_𝑓𝑥_{𝑓𝑓∈Γ}=𝑥_𝑓∈𝑈⟺𝑥_𝑓(𝑠0)∈𝑉⟺𝑥_𝑠0∈𝑊⟺𝑥∈_𝑠∈𝑆𝐵_𝑠⊆_𝑠∈𝑆𝑋_𝑠=𝑋. (4.13)
Since the map ℎ is bijection and 𝑝_𝑓⁻¹(𝑈)=ℎℎ⁻¹𝑝_𝑓⁻¹(𝑈)=ℎ_𝑠∈𝑆𝐵_𝑠 (4.14) for any subbase subset ∏_𝑠∈𝑆𝐵_𝑠⊆𝑋, the map ℎ is open.

In the case 𝜅=𝜔 we have well-known results that product of I-favorable space is I-favorable space (see [1] or [2]).

Corollary 4.2. Every I-favorable space is stable under any product.

If 𝐷 is a set and 𝜅 is cardinal number then we denote ⋃_𝛼<𝜅𝐷^𝛼 by 𝐷^<𝜅.

The following result probably is known but we give a proof for the sake of completeness.

Theorem 4.3. 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^<𝑘, where 𝜅𝜏=^<𝜅𝜅,forregular𝜅,_𝜅,otherwise. (4.15)

Proof. Assume that 𝜅 is regular cardinal. Let 𝐴∈[𝑇]^𝜅 and let 𝑓_𝛿∶⋃_𝛼<𝜅𝑇^𝛼→𝑇 for 𝛿<𝜅^<𝜅. Let 𝐴₀=𝐴. Assume that we have defined 𝐴_𝛼 for 𝛼<𝛽 such that |𝐴_𝛼|≤𝜅^|𝛼|. Put 𝐴_𝛽=_𝛼<𝛽𝐴_𝛼∪𝑓_𝛿(𝐶)∶𝐶∈_𝛼<𝛽𝐴_𝛼^<𝛽,𝛿<𝜅^|𝛽|. (4.16) Calculate the size of the set 𝐴_𝛽: 𝐴_𝛽≤_𝛼<𝛽𝐴_𝛼𝜅^|𝛽|_𝛼<𝛽𝐴_𝛼^<𝛽≤𝜅^|𝛽|𝜅^|𝛽||𝛽|≤𝜅^|𝛽|. (4.17)
Let 𝐵=⋃_𝛽<𝜅𝐴_𝛽, so we get |𝐵|≤𝜅^<𝜅. Fix a sequence ⟨𝑏_𝛼∶𝛼<𝛽⟩⊆𝐵 and 𝑓_𝛾. Since cf(𝜅)=𝜅 there exists 𝛿<𝜅 such that 𝐶={𝑏_𝛼∶𝛼<𝛽}⊆𝐴_𝛿 and 𝑓_𝛾(𝐶)∈𝐴_𝜎+1 for some 𝜎<𝜅.
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 4.4. If 𝑋 belongs to the class 𝒞_𝜅 then 𝑐(𝑋)≤𝜅.

Proof. If 𝑋∈𝒞_𝜅 then by Theorems 3.3 and Proposition 2.2 we get 𝑐(𝑋)≤𝜇(𝑋)≤𝜅.

We apply some facts from the paper [3]. Let 𝒫 be a family of open subset of topological space 𝑋 and 𝑥,𝑦∈𝑋. We say that 𝑥∼_𝒫𝑦 if and only if 𝑥∈𝑉⇔𝑦∈𝑉 for every 𝑉∈𝒫. The family of all sets [𝑥]_𝒫={𝑦∶𝑦∼_𝒫𝑥} 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 [3]: 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.

Notice that if 𝒫 has a property ∀(𝑊∈𝒫)∃𝑉_𝑛∶𝑛<𝜔⊆𝒫∃𝑈_𝑛∶𝑛<𝜔⊆𝒫,𝑊=_𝑛<𝜔𝑈_𝑛,∀(𝑛<𝜔)𝑈_𝑛⊆𝑋⧵𝑉_𝑛⊆𝑈_𝑛+1, seq then ⋃𝒫=𝑋 and by [3, Lemma 3] the topology 𝒯_𝒫 is Hausdorff. Moreover if 𝒫 is closed under finite intersection then by [3, Lemma 4] the topology 𝒯_𝒫 is regular. Theorem 5 and Lemma 9 [3] yeild.

Theorem 4.5. If 𝒫 is a set of open subset of topological space 𝑋 such that (1)is closed under 𝜅-winning strategy, finite union and 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 𝜔₁ we get only 𝑋∈𝒞_𝜔1𝜔.

Theorem 4.6. Each Tichonov space 𝑋 with 𝜇(𝑋)=𝜅 can be dense embedded into inverse limit of a system {𝑋_𝜎,𝜋^𝜎_𝜚,Σ}, where all bonding map are skeletal, indexing set Σ is 𝜏-complete each 𝑋_𝜎 is Tichonov space with 𝑤(𝑋_𝜎)≤𝜏 and 𝜅𝜏=^<𝜅𝜅,forregular𝜅,_𝜅,otherwise. (4.18)

Proof. Let ℬ be a 𝜋-base for topological space 𝑋 consisting of cozero sets 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 follows: 𝑔({𝐵₀,𝐵₁,…,𝐵_𝑛})=𝐵₀∩𝐵₁∩⋯∩𝐵_𝑛 and ℎ({𝐵₀,𝐵₁,…,𝐵_𝑛})=𝐵₀∪𝐵₁∪⋯∪𝐵_𝑛. For each cozero set 𝑉∈ℬ fix a continuous function 𝑓_𝑉∶𝑋→[0,1] such that 𝑉=𝑓_𝑉⁻¹((0,1]). Put 𝜎_2𝑛(𝑉)=𝑓_𝑉⁻¹((1/𝑛,1]) and 𝜎_2𝑛+1(𝑉)=𝑓_𝑉⁻¹([0,1/𝑛)). By Theorem 4.3 for each ℛ∈[ℬ]^𝜅 and all functions ℎ,𝑔,𝜎_𝑛,𝜎 there is subset 𝒫⊆ℬ such that (1)|𝒫|≤𝜏, where 𝜅𝜏=^<𝜅𝜅,forregular𝜅,_𝜅,otherwise, (4.19)(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 4.5 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) and the (4). If Σ is directed by inclusion. It is easy to check that Σ is 𝜏-complete. Similar to [3, Theorem 11] we define a function 𝑓∶𝑋→𝑌 as follows 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, that is, 𝑓(𝑥)∈𝑌. It easy to see that 𝑓 is homeomorphism onto its image and 𝑓[𝑋] is dense in 𝑌, compare [3, proof of Theorem 11].

Theorem 4.6 suggests question.

Does each space 𝑋 belong to 𝒞_𝜇(𝑋)?

Fleissner [13] proved that there exists a space 𝑌 such that 𝑐(𝑌)=ℵ₀ and 𝑐(𝑌³)=ℵ₂. Hence, we get 𝜇(𝑌)=ℵ₁, by Theorem 3.3 and Corollary 4.2. Suppose that 𝑌∈𝒞_𝜇(𝑋) then 𝑐(𝑌³)≤ℵ₁, by Theorem 4.4, a contradiction.

Corollary 4.7. If 𝑋 is topological space with 𝜇(𝑋)=𝜅 then 𝑐(𝑋^𝐼)≤𝜏 and 𝜅𝜏=^<𝜅𝜅,forregular𝜅,_𝜅,otherwise. (4.20)

Proof. By Theorem 4.3 we get 𝑋^𝐼∈𝒞_𝜏. Hence by Theorems 4.4 and 4.1 we have 𝑐(𝑋^𝐼)≤𝜏.

By above Corollary we get the following.

Corollary 4.8 (see [14, Kurepa]). If {𝑋_𝑠∶𝑠∈𝑆} is a family of topological spaces and 𝑐(𝑋_𝑠)≤𝜅 for each 𝑠∈𝑆, then 𝑐(∏{𝑋_𝑠∶𝑠∈𝑆})≤2^𝜅.

Acknowledgment

The author thanks the referee for careful reading and valuable suggestions.