On functional tightness of infinite products

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 $t\left( \prod_{\alpha\leq \kappa} X_\alpha \right)\leq \kappa$,where$t(X)$isthetightnessofaspace$X$. 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 $\lambda \leq 2^\kappa$or$\lambda$islessthanthefirstmeasurablecardinal,then$t_0\left( \prod_{\alpha<\lambda} X_\alpha \right)\leq \kappa$,where$t_0(X)$isthefunctionaltightnessofaspace$X$. 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.