Editing Covering space (section)

=== Local homeomorphism ===
Since a covering <math>\pi:E \rightarrow X</math> maps each of the disjoint open sets of <math>\pi^{-1}(U)</math> homeomorphically onto <math>U</math> it is a local homeomorphism, i.e. <math>\pi</math> is a continuous map and for every <math>e \in E</math> there exists an open neighborhood <math>V \subset E</math> of <math>e</math>, such that <math>\pi|_V : V \rightarrow \pi(V)</math> is a homeomorphism.

It follows that the covering space <math>E</math> and the base space <math>X</math> locally share the same properties.

* If <math>X</math> is a connected and [[non-orientable manifold]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> of degree <math>2</math>, whereby <math>\tilde X</math> is a connected and orientable manifold.{{r|Hatcher|p=234}}
* If <math>X</math> is a connected [[Lie group]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> which is also a [[Lie group homomorphism]] and <math>\tilde X := \{\gamma:\gamma \text{ is a path in X with  }\gamma(0)= \boldsymbol{1_X} \text{ modulo homotopy with fixed ends}\}</math> is a Lie group.<ref>{{Cite book|last=Kühnel |first=Wolfgang |title=Matrizen und Lie-Gruppen|date=6 December 2010 |publisher=Springer Fachmedien Wiesbaden GmbH|location=Stuttgart|isbn=978-3-8348-9905-7}}</ref>{{rp|p=174}}
* If <math>X</math> is a [[Graph theory#Graph|graph]], then it follows for a covering <math>\pi:E \rightarrow X</math> that <math>E</math> is also a graph.{{r|Hatcher|p=85}}
* If <math>X</math> is a connected [[manifold]], then there is a covering <math>\pi:\tilde X \rightarrow X</math>, whereby <math>\tilde X</math> is a connected and [[Simply connected space|simply connected]] manifold.{{r|Forster|p=32}}
* If <math>X</math> is a connected [[Riemann surface]], then there is a covering <math>\pi:\tilde X \rightarrow X</math> which is also a holomorphic map{{r|Forster|p=22}} and <math>\tilde X</math> is a connected and simply connected Riemann surface.{{r|Forster|p=32}}