Compact spaces and products of finite spaces

Abstract

It is shown that the compact spaces are precisely the extension closed subspaces of products of finite spaces, where a subspace is extension closed if every open cover of the subspace extends to an open cover of the entire space. Every closed subspace is extension closed, and for Hausdorff spaces the converse also holds. A single compact space U is constructed, such that every compact space is an extension closed subspace of a product of copies of U; this parallels precisely the property possessed by the unit interval with respect to compact Hausdorff spaces.