Tightness in product spaces

Abstract:

A product $\prod {X_i}$ of topological spaces ${X_i},i \in I$ will be said to preserve tightness if \[ \partial \left ( {\prod {X_i}} \right ) \leq \left | I \right | \cdot {\text {sup}}\left \{ {\partial \left ( {{X_i}} \right )\left | {i \in I} \right .} \right \}\] where $\partial \left ( X \right )$ denotes the tightness of $X$. We show $\prod {X_i}$ preserves tightness whenever each finite subproduct does. It is further shown that this is the case whenever each ${X_i}$ is a locally compact ${T_2}$-space, and whenever each ${X_i}$ is a locally Lindelöf ${T_3}$ $P$-space, extending 5.9 in [J].