Noetherian $π$-bases and Telgársky's Conjecture

TL;DR

This work analyzes Noetherian $π$-bases and their influence on topological games, proving every space admits a Noetherian $π$-base and introducing Noetherian tables to organize these bases. It proves a central theorem: if $X$ has a Noetherian $π$-base with a special table and NONEMPTY has a winning strategy in the Banach–Mazur game $BM(X)$, then NONEMPTY has a 2-tactic in $BM(X)$, generalizing Galvin’s theorem and tying into Telgársky’s conjecture. The authors develop a purification framework $pur(⋅)$ and several combinatorial conditions ($\star$, $\star\star$, $\dagger$, $\dagger\dagger$, $\bullet$) to capture the needed structure, and show that many natural bases (including Galvin bases) can be made to satisfy these hypotheses. They also provide concrete consequences (e.g., spaces with $πw(X)≤ω_1$ have the required base) and pose open questions on the exact relationships among base properties and their set-theoretic implications.

Abstract

We investigate Noetherian families and show that every topological space has a Noetherian $π$-base. We prove that if a topological space has some special Noetherian $π$-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 Telgársky's conjecture on this subject. One of our examples is that any space $X$ with $πw(X)\leq ω_1$ has this special Noetherian $π$-base. We pose some questions about this topic.

Theorems & Definitions (78)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Example 1
• Remark 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof