On functional tightness of infinite products

TL;DR

This paper extends Malykhin's theorem from tightness to functional tightness, showing that under certain set-theoretic conditions, the functional tightness of infinite products of compact spaces remains bounded, answering a question by Okunev.

Contribution

It proves a new theorem on the preservation of functional tightness in infinite products of compact spaces under specific cardinality constraints.

Findings

Functional tightness is preserved in large products of compact spaces under certain conditions.

The result holds if the index set's size is at most 2^kappa or below the first measurable cardinal.

Without measurable cardinals, functional tightness remains bounded in arbitrarily large products.

Abstract

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