Search results for "Topological game"
showing 4 items of 4 documents
Cardinal estimates involving the weak Lindelöf game
2021
AbstractWe 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 wi…
Cardinal inequalities involving the Hausdorff pseudocharacter
2023
We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $H\psi(X)$. This invariant has the property $\psi_c(X)\leq H\psi(X)\leq\chi(X)$ for a Hausdorff space $X$. We show the cardinality of a Hausdorff space $X$ is bounded by $2^{pwL_c(X)H\psi(X)}$, where $pwL_c(X)\leq L(X)$ and $pwL_c(X)\leq c(X)$. This generalizes results of Bella and Spadaro, as well as Hodel. We show additionally that if $X$ is a Hausdorff linearly Lindel\"of space such that $H\psi(X)=\omega$, then $|X|\le 2^\omega$, under the assumption that either $2^{<\mathfrak{c}}=\mathfrak{c}$ or $\mathfrak{c}<\aleph_\omega$. The following game-theoretic result is shown: i…
Infinite games and chain conditions
2015
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 $G_\delta$ 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 Lindel\"of property, every cover by $G_\delta$ sets has a continuum-sized subcollection whose union is $G_\delta$-dense.
Selective versions of chain condition-type properties
2015
We study selective and game-theoretic versions of properties like the ccc, weak Lindel\"ofness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal invariants of the continuum.