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

This paper is devoted 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 the categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal covering for non-homotopically Hausdorff spaces, equivalently existence of small covering spaces. Moreover, we prove that for semi-locally small loop spaces, X is a small loop space if and only if every covering of X is trivial if and only if is an indiscrete topological group.