On resolvability of products

Abstract

All spaces below are $T_0$ and crowded (i.e. have no isolated points). For $n \le \omega$ let $M(n)$ be the statement that there are $n$ measurable cardinals and $\Pi(n)$ ($\Pi^+(n)$) that there are $n+1$ (0-dimensional $T_2$) spaces whose product is irresolvable. We prove that $M(1),\,\Pi(1)$ and $\Pi^+(1)$ are equiconsistent. For $1 < n < \omega$ we show that $CON(M(n))$ implies $CON(\Pi^+(n))$. Finally, $CON(M(\omega))$ implies the consistency of having infinitely many crowded 0-dimensional $T_2$-spaces such that the product of any finitely many of them is irresolvable. These settle old problems of Malychin. Concerning an even older question of Ceder and Pearson, we show that the following are consistent modulo a measurable cardinal: (i) There is a 0-dimensional $T_2$ space $X$ with $\omega_2 \le \Delta(X) \le 2^{\omega_1}$ whose product with any countable space is not $\omega_2$-resolvable, hence not maximally resolvable. (ii) There is a monotonically normal space $X$ with $\Delta(X) = \aleph_\omega$ whose product with any countable space is not $\omega_1$-resolvable, hence not maximally resolvable. These significantly improve a result of Eckertson.