Baire extensions

Abstract

A space is Baire if every nonempty open set is of second category. Every T 1 space is shown to have a T 1 Baire extension, and those Hausdorff spaces with Hausdorff Baire extensions are characterized. If a Hausdorff space has a Hausdorff Baire extension, then its Fomin H-closed extension is a Baire space. A problem by Mioduszewski issolved by giving an example of a minimal Hausdorff space that is the countable union of minimal Hausdorff, nowhere dense subsets. Every regular (not necessarily T 1) space is shown to have a regular Baire extension if and only if every T 3 space has a T 3 Baire extension. Every completely regular (not necessarily T 1) space is shown to have a completely regular Baire extension.