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

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, 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.