A note on 𝛼-compact spaces

Abstract

For an infinite cardinal α \alpha , m ( α ) m(\alpha ) denotes the least measurable cardinal, if one exists, not less than α \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 ∈ I } \{ {X_i}:i \in I\} be a collection of spaces each having at least two elements. Then the k k -box product ( ∏ X i ) k {(\prod {X_i})_k} is α \alpha -compact if and only if either X i {X_i} is α \alpha -compact for each i ∈ I i \in I and k ⩽ m ( α ) k \leqslant m(\alpha ) or | I | > m ( α ) \left | I \right | > m(\alpha ) .