D-20 - H-Closed Spaces

Abstract

A result taught in a first course in topology is that a compact subspace of a Hausdorff space is closed. A Hausdorff space with the property of being closed in every Hausdorff space containing it as a subspace is called Hausdorff-closed (H-closed). H-closed spaces were introduced in 1924 by Alexandroff and Urysohn. They produced an example of an H-closed space that is not compact, showed that a regular H-closed space is compact, characterized a Hausdorff space as H-closed precisely when every open cover has a finite subfamily whose union is dense, and posed the question as to which Hausdorff spaces can be densely embedded in an H-closed space. These spaces enjoy many of the same properties of compact Hausdorff spaces. A compact Hausdorff space has no strictly coarser Hausdorff topology, that is, it is minimal Hausdorff. A Hausdorff space is a Katĕtov space if it has a coarser minimal Hausdorff topology. A space is Katĕtov if it is the remainder of an H-closed extension of a discrete space. This shows that every Katĕtov space is an H-set in some H-closed space.