On functional tightness of infinite products

Abstract

A classical theorem of Malykhin says that if $\{X_α:α\leqκ\}$ is a family of compact spaces such that $t(X_α)\leq κ$, for every $α\leqκ$, then $t\left( \prod_{α\leq κ} X_α\right)\leq κ$, where $t(X)$ is the tightness of a space $X$. In this paper we prove the following counterpart of Malykhin's theorem for functional tightness: Let $\{X_α:α<λ\}$ be a family of compact spaces such that $t_0(X_α)\leq κ$ for every $α<λ$. If $λ\leq 2^κ$ or $λ$ is less than the first measurable cardinal, then $t_0\left( \prod_{α<λ} X_α\right)\leq κ$, where $t_0(X)$ is the functional tightness of a space $X$. In particular, if there are no measurable cardinals, then the functional tightness is preserved by arbitrarily large products of compacta. Our result answers a question posed by Okunev.