A characterization of submetacompactness in terms of products

Abstract:

A space $X$ is said to be suborthocompact if for every open cover $\mathcal {U}$ of $X$ there is a sequence $\{ {\mathcal {V}_n}\}$ of open refinements of $\mathcal {U}$ such that for each $x \in X$ there is some $n \in \omega$ such that $\cap \{ V \in {\mathcal {V}_n}:x \in V\}$ is a neighborhood of $x$ in $X$. It is proved that a Tychonoff space $X$ is submetacompact if and only if the product $X \times \beta X$ is suborthocompact.