# The functional tightness of infinite products

The functional tightness $t_0(X)$ of a space $X$ is a cardinal invariant related to both the tightness $t(X)$ and the density character $d(X)$ of $X$. While the tightness $t(X)$ measures the minimal cardinality of sets required to determine the topology of $X$, the functional tightness

measures the minimal size of sets required to guarantee the continuity of real-valued functions on $X$.

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 my talk I will prove the following counterpart of Malykhin's theorem for functional tightness:

Let $\{X_\alpha:\alpha< \lambda\}$ be a family of compact spaces such that $t_0(X_\alpha)\leq \kappa$. 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 the functional tightness is preserved by arbitrarily large products of compacta.