Noetherian $\pi$-bases and Telg\'arsky's Conjecture

Abstract

We investigate Noetherian families and show that every topological space has a Noetherian $\pi$-base. We prove that if a topological space has some special Noetherian $\pi$-bases, then NONEMPTY has a 2-tactic in the Banach-Mazur game on a space $X$, denoted as $BM(X)$, whenever NONEMPTY has a winning strategy in BM(X). This result encompasses an important theorem of Galvin in this context and is related to Telg\'arsky's conjecture on this subject. One of our examples is that any space $X$ with $\pi w(X)\leq \omega_1$ has this special Noetherian $\pi$-base. We pose some questions about this topic.