Editing Covering space (section)

=== Existence ===
A universal covering does not always exist. The following theorem guarantees its existence for a certain class of base spaces.

Let <math>X</math> be a connected, [[Locally simply connected space|locally simply connected]] topological space. Then, there exists a universal covering <math>p:\tilde X \rightarrow X.</math>

The set <math>\tilde X</math> is defined as <math>\tilde X = \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{homotopy with fixed ends},</math> where <math>x_0 \in X</math> is any chosen base point. The map <math>p:\tilde X \rightarrow X</math> is defined by <math>p([\gamma])=\gamma(1).</math>{{r|Hatcher|p=64}}

The [[topology]] on <math>\tilde X</math> is constructed as follows: Let <math>\gamma:I \rightarrow X</math> be a path with <math>\gamma(0)=x_0.</math> Let <math>U</math> be a simply connected neighborhood of the endpoint <math>x=\gamma(1).</math> Then, for every <math>y \in U,</math> there is a [[path (topology)|path]] <math>\sigma_y</math> inside <math>U</math> from <math>x</math> to <math>y</math> that is unique up to [[homotopy]]. Now consider the set <math>\tilde U=\{\gamma\sigma_y:y \in U \}/\text{homotopy with fixed ends}.</math> The restriction <math>p|_{\tilde U}: \tilde U \rightarrow U</math> with <math>p([\gamma\sigma_y])=\gamma\sigma_y(1)=y</math> is a bijection and <math>\tilde U</math> can be equipped with the [[final topology]] of <math>p|_{\tilde U}.</math>{{explain|date=December 2024|reason=How do these topologies on the tilde-U combine into one on tilde-X?}}

The fundamental group <math>\pi_{1}(X,x_0) = \Gamma</math> acts [[Free group action|freely]] on <math>\tilde X</math> by <math>([\gamma],[\tilde x]) \mapsto [\gamma\tilde x],</math> and the orbit space <math>\Gamma \backslash \tilde X</math> is homeomorphic to <math>X</math> through the map <math>[\Gamma \tilde x]\mapsto\tilde x(1).</math>