Telgársky’s conjecture may fail

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