Meager Sets, Games and Singular Cardinals (original) (raw)
2018, arXiv (Cornell University)
Related papers
Infinite games and cardinal properties of topological spaces
Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the cardinality of weakly Lindel\"of first-countable regular spaces and answer a question recently asked by Babinkostova, Pansera and Scheepers. In the second part of the paper we study a game-theoretic version of cellularity, a special case of which has been introduced by Aurichi. We obtain a game-theoretic proof of Shapirovskii's bound for the number of regular open sets in an (almost) regular space and give a partial answer to a natural question about the productivity of a game strengthening of the countable chain condition that was introduced by Aurichi. As a final application of our results we prove that the Hajnal-Juh\'asz bound for the cardinality of a first-countable ccc Hausdorff space is true for almost regular (non-Hausdorff) spaces.
Cardinal estimates involving the weak Lindelöf game
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2021
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)G1ω1(O,OD)(theweakLindelo¨fgameoflengthG 1 ω 1 ( O , O D ) (the weak Lindelöf game of lengthG1ω1(O,OD)(theweakLindelo¨fgameoflength\omega _1ω1)thenXhascardinalityatmostcontinuum.ThismaybeconsideredapartialanswertoanoldquestionofBell,GinsburgandWoods.ItisalsothebestresultofthiskindsincethereareHausdorfffirst−countablespacesofarbitrarilylargecardinalitywhereplayerIIhasawinningstrategyevenintheweakLindelo¨fgameofcountablelength.Wealsotackletheproblemoffindingaboundonthecardinalityofafirst−countablespacewhereplayerIIhasawinningstrategyinthegameω 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 gameω1)thenXhascardinalityatmostcontinuum.ThismaybeconsideredapartialanswertoanoldquestionofBell,GinsburgandWoods.ItisalsothebestresultofthiskindsincethereareHausdorfffirst−countablespacesofarbitrarilylargecardinalitywhereplayerIIhasawinningstrategyevenintheweakLindelo¨fgameofcountablelength.Wealsotackletheproblemoffindingaboundonthecardinalityofafirst−countablespacewhereplayerIIhasawinningstrategyinthegameG^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)Gfinω1(O,OD),providingsomepartialanswerstoit.WefinishbyconstructinganexampleofacompactspacewhereplayerIIdoesnothaveawinningstrategyintheweakLindelo¨fgameoflengthG 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 lengthGfinω1(O,OD),providingsomepartialanswerstoit.WefinishbyconstructinganexampleofacompactspacewhereplayerIIdoesnothaveawinningstrategyintheweakLindelo¨fgameoflength\omega _1$$ ω 1 .
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.