Coloring Cantor sets and resolvability of pseudocompact spaces

TL;DR

This paper proves that certain feebly compact, $ ext{pi}$-regular spaces are $ ext{mu}$-resolvable under conditions involving colorings of Baire spaces, extending previous results in the field.

Contribution

It establishes a new link between coloring properties of Baire spaces and the resolvability of feebly compact $ ext{pi}$-regular spaces, improving earlier theorems.

Findings

Feebly compact $ ext{pi}$-regular spaces are $ ext{mu}$-resolvable under certain coloring conditions.

The result generalizes and strengthens previous theorems by van Mill and Ortiz-Castillo and Tomita.

The paper connects combinatorial coloring properties with topological resolvability.

Abstract

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X\,$ {\em $\pi$-regular} if it is Hausdorff and for every non-empty open set $U$ in $X$ there is a non-empty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called {\em feebly compact} if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \widehat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. contains $\mu$ pairwise disjoint dense subsets. (Here $\widehat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill , resp. Ortiz-Castillo and Tomita.