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

TL;DR

This paper explores Noetherian $ ext{pi}$-bases in topological spaces, linking their properties to Telgársky's conjecture and the Banach-Mazur game, and extends known results with new examples and questions.

Contribution

It proves that every space has a Noetherian $ ext{pi}$-base and identifies conditions under which NONEMPTY has a 2-tactic in the Banach-Mazur game, relating to Telgársky's conjecture.

Findings

Every topological space has a Noetherian $ ext{pi}$-base.

Spaces with certain Noetherian $ ext{pi}$-bases allow NONEMPTY to have a 2-tactic in BM(X).

Spaces with $ ext{pi}w(X) extless ext{omega}_1$ have the special Noetherian $ ext{pi}$-base.

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.