We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game G{ω_1}_1(O,O_D) (the weak Lindelöf game of length ω1) 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^{ω_1}_{fin}(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 ω_1.
Aurichi L., Bella A., Spadaro S. (2022). Cardinal estimates involving the weak Lindelöf game. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES. SERIE A, MATEMÁTICAS, 116 [10.1007/s13398-021-01141-0].
Cardinal estimates involving the weak Lindelöf game
Spadaro S.
2022-01-01
Abstract
We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game G{ω_1}_1(O,O_D) (the weak Lindelöf game of length ω1) 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^{ω_1}_{fin}(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 ω_1.
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