Resolvability in products and squares

Abstract

Suppose $X$ and $Y$ are topological spaces, $|X| = Δ(X)$ and $|Y| = Δ(Y)$. We investigate resolvability of the product $X \times Y$. We prove that: I. If $|X| = |Y| = ω$ and $X,Y$ are Hausdorff, then $X \times Y$ is maximally resolvable; II. If $2^κ= κ^+$, $\{|X|, \mathrm{cf}|X|\} \cap \{κ, κ^+\} \ne \emptyset$ and $\mathrm{cf}|Y| = κ^+$, then the space $X \times Y$ is $κ^+$-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 $\mathrm{cf}|X| = ω$ and $\mathrm{cf}|Y| = \mathrm{cf}(\frak{c})$, then the space $X \times Y$ is $ω$-resolvable. If, moreover, $\mathrm{cf}(\frak{c}) = ω_1$, then the space $X \times Y$ is $ω_1$-resolvable.