Pseudocompactness properties

Abstract:

A topological extension property is a class of Tychonoff spaces $\mathcal {P}$ which is closed hereditary, closed under formation of topological products and contains all compact spaces. If $X$ is Tychonoff and $\mathcal {P}$ is an extension property, there is a space $\mathcal {P}X$ such that $X \subseteq \mathcal {P}X \subseteq \beta X,\;\mathcal {P}X \in \mathcal {P}$ and if $f:X \to Y$ where $Y \in \mathcal {P}$ then $f$ admits a continuous extension to $\mathcal {P}X$. A space $X$ is called $\mathcal {P}$-pseudocompact if $\mathcal {P}X = \beta X$. In this note it is shown that if $\mathcal {P}$ is an extension property which contains the real line (e.g., the class of realcompact spaces), $X$ is $\mathcal {P}$-pseudocompact and $Y$ is compact, then $X \times Y$ is $\mathcal {P}$-pseudocompact. An example is given of an extension property $\mathcal {P}$, a $\mathcal {P}$-pseudocompact space $X$ and a compact space $Y$ such that $X \times Y$ is not $\mathcal {P}$-pseudocompact.