On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane

G. R. Conner¹ ; J. W. Lamoreaux²

¹Mathematics Department Brigham Young University Provo, UT 84602, U.S.A.
²Mathematics Department Brigham Young University Provo, UT 84602

Fundamenta Mathematicae, Tome 187 (2005) no. 2, pp. 95-110

Résumé

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, ${\mathbb E}^2$, admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.

Détail

DOI : 10.4064/fm187-2-1

Keywords: prove several results concerning existence universal covering spaces separable metric spaces begin define several homotopy theoretic conditions which prove equivalent existence universal covering space these equivalences prove every connected locally path connected separable metric space whose fundamental group group admits universal covering space application these results prove main result article which states connected locally path connected subset euclidean plane mathbb admits universal covering space only its fundamental group only its fundamental group countable