On functional tightness of infinite products

Abstract

A classical theorem of Malykhin says that if $\{X_{\alpha }:\alpha \le \kappa \}$ is a family of compact spaces such that $t(X_{\alpha })\le \kappa$, for every $\alpha \le \kappa$, then $t\left({\prod }_{\alpha \le \kappa }{X}_{\alpha }\right)\le \kappa$, 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 $\{X_{\alpha }:\alpha <\lambda \}$ be a family of compact spaces such that $t_0(X_{\alpha })\le \kappa$ for every $\alpha <\lambda$. If $\lambda \le 2^{\kappa }$ or $\lambda$ is less than the first measurable cardinal, then $t_0\left({\prod }_{\alpha <\lambda }{X}_{\alpha }\right)\le \kappa$, 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.