Resolvability in products and squares

Abstract

Suppose $X$ and $Y$ are topological spaces, $|X|=\hat{I}”(X)$ and $|Y|=\hat{I}”(Y)$. We investigate resolvability of the product $X\times Y$. We prove that: I. If $|X|=|Y|=\ddot{I}‰$ and $X,Y$ are Hausdorff, then $X\times Y$ is maximally resolvable; II. If $2^{\hat{I}}º=\hat{I}{º}^{+}$, $\{|X|,\mathrm{cf}|X|\}\cap \{\hat{I}º,\hat{I}{º}^{+}\}\ne \varnothing$ and $\mathrm{cf}|Y|=\hat{I}{º}^{+}$, then the space $X\times Y$ is $\hat{I}{º}^{+}$-resolvable. In particular, under GCH the space $X^2$ is $\mathrm{cf}|X|$-resolvable whenever $\mathrm{cf}|X|$ is an isolated cardinal; III. ($\mathfrak{r}=\mathfrak{c}$) If $\mathrm{cf}|X|=\ddot{I}‰$ and $\mathrm{cf}|Y|=\mathrm{cf}(\mathfrak{c})$, then the space $X\times Y$ is $\ddot{I}‰$-resolvable. If, moreover, $\mathrm{cf}(\mathfrak{c})=\ddot{I}{‰}_1$, then the space $X\times Y$ is $\ddot{I}{‰}_1$-resolvable.

10 pages. Minor changes