Resolvability in products and squares

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