Resolvability in products and squares

TL;DR

This paper studies the resolvability properties of product spaces, establishing conditions under which such products are maximally resolvable or have specific resolvability levels, depending on the spaces' cardinalities and set-theoretic assumptions.

Contribution

It provides new results on the resolvability of product spaces with various cardinality and set-theoretic conditions, extending previous understanding.

Findings

Product of two Hausdorff spaces with countable size is maximally resolvable.

Under certain set-theoretic assumptions, the product space is -resolvable.

Resolvability levels depend on the cofinality of the spaces' cardinalities.

Abstract

Suppose $X$ and $Y$ are topological spaces, $|X| = \Delta(X)$ and $|Y| = \Delta(Y)$. We investigate resolvability of the product $X \times Y$. We prove that: I. If $|X| = |Y| = \omega$ and $X,Y$ are Hausdorff, then $X \times Y$ is maximally resolvable; II. If $2^\kappa = \kappa^+$, $\{|X|, \mathrm{cf}|X|\} \cap \{\kappa, \kappa^+\} \ne \emptyset$ and $\mathrm{cf}|Y| = \kappa^+$, then the space $X \times Y$ is $\kappa^+$-resolvable. In particular, under GCH the space $X^2$ is $\mathrm{cf}|X|$-resolvable whenever $\mathrm{cf}|X|$ is an isolated cardinal; III. ($\frak{r} = \frak{c}$) If $\mathrm{cf}|X| = \omega$ and $\mathrm{cf}|Y| = \mathrm{cf}(\frak{c})$, then the space $X \times Y$ is $\omega$-resolvable. If, moreover, $\mathrm{cf}(\frak{c}) = \omega_1$, then the space $X \times Y$ is $\omega_1$-resolvable.