On resolvability of products

TL;DR

This paper explores the resolvability of products of certain topological spaces under various set-theoretic assumptions, establishing equiconsistency results and settling longstanding open problems in topology.

Contribution

It proves the equiconsistency of the existence of irresolvable products of spaces with the existence of measurable cardinals, and addresses old questions by showing consistency results involving resolvability.

Findings

Equiconsistency of $M(1)$, $ ext{Pi}(1)$, and $ ext{Pi}^+(1)$.

Implication from $CON(M(n))$ to $CON( ext{Pi}^+(n))$ for $1<n< ext{omega}$.

Consistency of spaces with large dispersion character whose products are not resolvable.

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.