On a game theoretic cardinality bound

The main purpose of the paper is the proof of a cardinal inequality for a space with points $G_\delta$, obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall

Topological games and productively countably tight spaces

The two main results of this work are the following: if a space $X$ is such that player II has a winning strategy in the game \gone(\Omega_x, \Omega_x) for every $x \in X$, then $X$ is productively countably tight. On the other hand, if a space is productively countably tight, then \sone(\Omega_x, \Omega_x) holds for every $x \in X$. With these results, several other results follow, using some characterizations made by Uspenskii and Scheepers

Lindelof spaces which are indestructible, productive, or D

We discuss relationships in Lindelof spaces among the properties "indestructible", "productive", "D", and related properties

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}

Cardinal estimates involving the weak Lindelöf game

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 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