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

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

2010-11-07

arxiv.org/abs/1011.1611v1

Topology and its Applications, 158 (2011) 803-809

Mathematics

Algebraic Topology

7 pages

Scientific paper

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.

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