Coloring Cantor sets and resolvability of pseudocompact spaces

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 $\lambda <\hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ 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.

8 pages