We prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy-theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalences to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article, which states that a connected, locally path connected subset of the Euclidean plane, 𝔼², admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-1,
author = {G. R. Conner and J. W. Lamoreaux},
title = {On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane},
journal = {Fundamenta Mathematicae},
volume = {185},
year = {2005},
pages = {95-110},
zbl = {1092.57001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-1}
}
G. R. Conner; J. W. Lamoreaux. On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane. Fundamenta Mathematicae, Tome 185 (2005) pp. 95-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm187-2-1/