Telgársky’s conjecture states that for each k ∈ ℕ, there is a topological space X_k such that in the Banach-Mazur game on X_k, 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∣.

Telgársky猜想指出，每个ķ ∈ℕ，有一个拓扑空间X _ķ使得在巴拿赫-马祖尔游戏X _ķ，玩家非空的有一个获胜（ķ + 1）-tactic，但没有获胜ķ -tactic。我们证明该陈述始终是错误的。

更具体地说，假设GCH +□，我们证明如果非空在T ₃空间X上对Banach-Mazur游戏有获胜策略，则她有2获胜策略。证明使用了加尔文（Galvin）的编码参数，因此，如果X具有具有某些良好属性的π基，则非空值能够在她的对手的每一对连续动作中对与比赛前的比赛打法有关的所有基本信息进行编码。目前的举动。我们的证明表明，在GCH +□下，每个T ₃空间都有一个足够好的π基，可以启用这种编码策略。

翻译成偏序集的语言，我们真正显示的是GCH +□意味着下面的语句，也就是等同于“好”的存在π上述-bases：∇：每分离性偏序ℙ与κ - cc包含一个密集的子坐位\（\ mathbb {D} \），使得\（\ left | {\ left \ {{q \ in \ mathbb {D}：p \; {\ rm {extends}} \\; q} \右\}} \右| <\卡帕\）对每个p ∈ℙ。

我们证明该语句独立于ZFC：尽管它在GCH +□下成立，但即使\（\ mathfrak {b}> {\ aleph _1} \）对于ccc姿势也是错误的。我们还表明，如果\（\ left | \ mathbb {P} \ right | <{\ aleph _ \ omega} \），则∇-for-ℙ是GCH保持在∣ℙ∣以下的结果。