Selective versions of chain condition-type properties

2021

RACSAM

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .

Section: The Weak Lindelöf Game and Cardinalitymentioning

confidence: 93%

Section: Question 31 Let X Be a First-countable Regular Space Where Player Ii Has A Winning Strategy Inmentioning

confidence: 99%

Cardinal estimates involving the weak Lindelöf game

Aurichi

Bella

2021

RACSAM

“…A game-theoretic version of the weak Lindelöf property can be obtained by considering the game G In [4] we proved that the weak Lindelöf game is the dual of the open-picking game. It will be convenient to exploit this duality in the proof of our partial solution to Arhangel'skii's problem.…”

Section: Arhangel'skii's Problem About G δ Covers In Compact Spacesmentioning

confidence: 94%

“…Proof. We prove only the direct implication of (2), because it's the only one we will need in our proof of Theorem 2.3 below, and we refer the reader to [4] for the other implications.…”

Section: Arhangel'skii's Problem About G δ Covers In Compact Spacesmentioning

confidence: 99%

“…Infinite games have been exploited in recent years to give partial answers to various important problems in General Topology, including van Douwen's D-space problem (see [1]), Arhangel'skii's problem on the cardinality of Lindelöf spaces with points G δ (see [14], [2]) and Bell, Ginsburg and Woods's problem on the cardinality of weakly Lindelöf first-countable regular spaces (see [6], [3], [4]). We use them to give partial ZFC answers to Suslin's Problem and Arhangel'skii's question of whether in every compact space, a cover by G δ sets has a continuum-sized subfamily with a G δ -dense union.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Infinite games and chain conditions

2016

Fund. Math.

Abstract. We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on G δ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by G δ sets has a continuum-sized subcollection whose union is G δ -dense. IntroductionInfinite games have been exploited in recent years to give partial answers to various important problems in General Topology, including van Douwen's D-space problem (see [1]), Arhangel'skii's problem on the cardinality of Lindelöf spaces with points G δ (see [14], [2]) and Bell, Ginsburg and Woods's problem on the cardinality of weakly Lindelöf first-countable regular spaces (see [6], [3], [4]). We use them to give partial ZFC answers to Suslin's Problem and Arhangel'skii's question of whether in every compact space, a cover by G δ sets has a continuum-sized subfamily with a G δ -dense union.It was already known to Cantor that the real line is the unique complete dense linear order without endpoints which is separable. In the first issue of Fundamenta Mathematicae, Suslin asked whether separability could be replaced with the countable chain condition in this result. Any counterexample to this assertion came to be known as a Suslin Line. The problem turned out to be independent of the usual axioms of ZFC: under MA ω 1 there are no Suslin Lines, and thus the answer to Suslin's question is yes. However, Suslin Lines can be found in certain models of set theory, for example under V = L. Various mathematicians wondered whether there was a natural strengthening of the ccc which implied a positive answer to Suslin's Problem. For example, Knaster proved that every ordered continuum with the Knaster property is separable and Shapirovskii proved that every compact space with countable tightness and Shanin's condition is separable (see [15]).2000 Mathematics Subject Classification. Primary: 54A25, 91A44; Secondary: 54F05, 54G10.

Cardinal inequalities involving the Hausdorff pseudocharacter

Bella

Carlson

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

2023

We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $$H\psi (X)$$ H ψ ( X ) . This invariant has the property $$\psi _c(X)\le H\psi (X)\le \chi (X)$$ ψ c ( X ) ≤ H ψ ( X ) ≤ χ ( X ) for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by $$2^{pwL_c(X)H\psi (X)}$$ 2 p w L c ( X ) H ψ ( X ) , where $$pwL_c(X)\le L(X)$$ p w L c ( X ) ≤ L ( X ) and $$pwL_c(X)\le c(X)$$ p w L c ( X ) ≤ c ( X ) . This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if X is a Hausdorff linearly Lindelöf space such that $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω , under the assumption that either $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c or $${\mathfrak {c}}<\aleph _\omega $$ c < ℵ ω . The following game-theoretic result is shown: if X is a regular space such that player two has a winning strategy in the game $$G^{\kappa }_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 κ ( O , O D ) , $$H \psi (X) < \kappa $$ H ψ ( X ) < κ and $$\chi (X) \le 2^{<\kappa }$$ χ ( X ) ≤ 2 < κ , then $$|X| \le 2^{<\kappa }$$ | X | ≤ 2 < κ . This improves a result of Aurichi, Bella, and Spadaro. Generalizing a result for first-countable spaces, we demonstrate that if X is a Hausdorff almost discretely Lindelöf space satisfying $$H\psi (X)=\omega $$ H ψ ( X ) = ω , then $$|X|\le 2^\omega $$ | X | ≤ 2 ω under the assumption $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ 2 < c = c . Finally, we show $$|X|\le 2^{wL(X)H\psi (X)}$$ | X | ≤ 2 w L ( X ) H ψ ( X ) if X is a Hausdorff space with a $$\pi $$ π -base with elements with compact closures. This is a variation of a result of Bella, Carlson, and Gotchev.