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

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Conner G., Lamoreaux J.

Fundamenta Mathematicae

2005

tom 187

nr 2

95-110

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Conner G.

1 Lamoreaux J.

1 2005

Wyniki wyszukiwania

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