A note on $\alpha$-compact spaces

Abstract:

For an infinite cardinal $\alpha$, $m(\alpha )$ denotes the least measurable cardinal, if one exists, not less than $\alpha$. We give easy proofs of generalizations of some results on realcompact spaces. Among these we prove the following generalization of a theorem of A. Kato. Let $\{ {X_i}:i \in I\}$ be a collection of spaces each having at least two elements. Then the $k$-box product ${(\prod {X_i})_k}$ is $\alpha$-compact if and only if either ${X_i}$ is $\alpha$-compact for each $i \in I$ and $k \leqslant m(\alpha )$ or $\left | I \right | < m(\alpha )$.