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 X̃/H → X, where H is a subgroup of the fundamental group of X and X̃ equipped with the topology introduced in Spanier's Algebraic Topology. The projection X̃/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on X̃ called the lasso topology. Then the fundamental group π₁(X) as a subspace of X̃ with the lasso topology becomes a topological group. Also, one has a characterization of X̃/H → X having the unique path lifting property if H is a normal subgroup of π₁(X). Namely, H must be closed in π₁(X) with the lasso topology. Such groups include π(𝓤,x₀) (𝓤 being an open cover of X) and the kernel of the natural homomorphism π₁(X,x₀) → π̌₁(X,x₀).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-2,
author = {N. Brodskiy and J. Dydak and B. Labuz and A. Mitra},
title = {Covering maps for locally path-connected spaces},
journal = {Fundamenta Mathematicae},
volume = {219},
year = {2012},
pages = {13-46},
zbl = {1260.55013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-2}
}
N. Brodskiy; J. Dydak; B. Labuz; A. Mitra. Covering maps for locally path-connected spaces. Fundamenta Mathematicae, Tome 219 (2012) pp. 13-46. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm218-1-2/