Small loop spaces and covering theory of non-homotopically Hausdorff spaces

TL;DR

This paper investigates the covering theory of non-homotopically Hausdorff spaces, introducing small coverings and semi-locally small loop spaces, and characterizing the existence of universal covers in this context.

Contribution

It introduces the notions of small covering and semi-locally small loop spaces, providing criteria for the existence of universal covers in non-homotopically Hausdorff spaces.

Findings

Every small covering is the universal covering in a categorical sense.

Existence of universal cover characterized by semi-locally small loop space condition.

For semi-locally small loop spaces, small loop space condition relates to trivial covers and indiscrete topological fundamental group.

Abstract

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of $X$ is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, $X$ is a small loop space if and only if every cover of $X$ is trivial if and only if $\pi_1^{top}(X)$ is an indiscrete topological group.