The Baire property in uniform spaces: a survey

TL;DR

This survey explores conditions under which complete uniform spaces possess the Baire property, extending classical results from metric spaces to more general uniform spaces.

Contribution

It investigates the applicability of the Baire property in uniform spaces under various conditions like countable compactness and pseudocompleteness.

Findings

Complete uniform spaces do not always satisfy the Baire property.

Certain conditions such as pseudocompactness influence the Baire property in uniform spaces.

The survey clarifies when uniform spaces exhibit Baire space characteristics.

Abstract

The Baire category theorem states that every complete pseudometric space is a Baire space. There are some results in metric spaces which have their analogue in uniform spaces, however this is not one of them. Nonetheless, since the Baire property is always desirable, we decide to explore some conditions, such as countable compacity, pseudocompacity and pseudocompleteness, and see in which circumstances a general complete uniform space satisfies the Baire property.