Infinite games and cardinal properties of topological spaces

Abstract

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 Lindelöf 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-Juhász bound for the cardinality of a first-countable ccc Hausdorff space is true for almost regular (non-Hausdorff) spaces.

Increasing chains and discrete reflection of cardinality

“…x ∈ D}. By Shapirovskii's bound for the number of regular open sets (see [17] or [20] or [6] for a game-theoretic proof) we have that ρ(X) ≤ πw(X) c(X) ≤ (2 κ ) κ = 2 κ . Moreover, since by Jones Lemma ρ(X) ≥ 2 |D| in every hereditarily normal space X, we must have |D| < κ and hence D ∈ M. Therefore D ∈ M. From |D| ≤ 2 c(X) we get that D ⊂ X ∩ M and thus p / ∈ D. This implies that there is x ∈ D such that p ∈ U x .…”