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 $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: 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. $X$ 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 J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].