Editing Covering space (section)

==== Properties ====
Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Let <math>H=p_{\#}(\pi_1(E))</math> be a [[subgroup]] of <math>\pi_1(X)</math>, then <math>p</math> is a normal covering iff <math>H</math> is a [[normal subgroup]] of <math>\pi_1(X)</math>.

If <math>p:E \rightarrow X</math> is a normal covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong \pi_1(X)/H</math>.

If <math>p:E \rightarrow X</math> is a path-connected covering and <math>H=p_{\#}(\pi_1(E))</math>, then <math>\operatorname{Deck}(p) \cong N(H)/H</math>, whereby <math>N(H)</math> is the [[normaliser]] of <math>H</math>.{{r|Hatcher|p=71}}

Let <math>E</math> be a topological space. A group <math>\Gamma</math> acts ''discontinuously'' on <math>E</math>, if every <math>e \in E</math> has an open neighborhood <math>V \subset E</math>  with <math>V \neq \empty</math>, such that for every <math>d_1, d_2 \in \Gamma
</math> with <math>d_1 V \cap d_2 V \neq \empty
</math> one has <math>d_1 = d_2</math>.

If a group <math>\Gamma</math> acts discontinuously on a topological space <math>E</math>, then the [[quotient map (topology)|quotient map]] <math>q: E \rightarrow \Gamma \backslash E </math> with <math>q(e)=\Gamma e</math> is a normal covering.{{r|Hatcher|p=72}} Hereby <math>\Gamma \backslash E = \{\Gamma e: e \in E\}</math> is the [[Quotient space (topology)|quotient space]] and <math>\Gamma e = \{\gamma(e):\gamma \in \Gamma\}</math> is the [[Orbit (group theory)|orbit]] of the group action.