Telgársky’s conjecture may fail

Abstract

Telgársky’s conjecture states that for each k ∈ ℕ, there is a topological space Xk such that in the Banach-Mazur game on Xk, the player nonempty has a winning (k + 1)-tactic but no winning k-tactic. We prove that this statement is consistently false.More specifically, we prove, assuming GCH+□, that if nonempty has a winning strategy for the Banach-Mazur game on a T3 space X, then she has a winning 2-tactic. The proof uses a coding argument due to Galvin, whereby if X has a π-base with certain nice properties, then nonempty is able to encode, in each consecutive pair of her opponent’s moves, all essential information about the play of the game before the current move. Our proof shows that under GCH + □, every T3 space has a sufficiently nice π-base that enables this coding strategy.Translated into the language of partially ordered sets, what we really show is that GCH + □ implies the following statement, which is equivalent to the existence of the “nice” π-bases mentioned above: ∇: Every separative poset ℙ with the κ-cc contains a dense sub-poset \(\mathbb{D}\) such that \(\left| {\left\{ {q \in \mathbb{D} :p\;{\rm{extends}}\;q} \right\}} \right| < \kappa \) for every p ∈ ℙ.We prove that this statement is independent of ZFC: while it holds under GCH + □, it is false even for ccc posets if \(\mathfrak{b} > {\aleph _1}\). We also show that if \(\left|\mathbb{P} \right| < {\aleph _\omega }\), then ∇-for-ℙ is a consequence of GCH holding below ∣ℙ∣.