Covering maps for locally path-connected spaces

Abstract

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow. If $X$ is path-connected, then every Peano covering map is equivalent to the projection $\tilde{X}/H\to X$, where $H$ is a subgroup of the fundamental group of $X$ and $\tilde{X}$ equipped with the basic topology. The projection $\tilde{X}/H\to X$ is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on $\tilde{X}$ for which one has a characterization of $\tilde{X}/H\to X$ having the unique path lifting property if $H$ is a normal subgroup of ${\pi }_1(X)$. Namely, $H$ must be closed in ${\pi }_1(X)$. Such groups include $\pi (\mathcal{U},x_0)$ ($\mathcal{U}$ being an open cover of $X$) and the kernel of the natural homomorphism from the fundamental group to the Cech fundamental group.

25 pages, several references added plus Proposition 4.12 and Example 4.13