Editing Covering space (section)

== Properties ==
=== 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}}

=== Factorisation ===
Let <math> X, Y</math> and <math>E</math> be path-connected, locally path-connected spaces, and <math>p,q</math> and <math>r</math> be continuous maps, such that the diagram 
[[File:Commutativ coverings.png|center|frameless]]
commutes.

* If <math>p</math> and <math>q</math> are coverings, so is <math>r</math>.
* If <math>p</math> and <math>r</math> are coverings, so is <math>q</math>.{{r|Munkres|p=485}}

=== Product of coverings ===
Let <math>X</math> and <math>X'</math> be topological spaces and <math>p:E \rightarrow X</math> and <math>p':E' \rightarrow X'</math> be coverings, then <math>p \times p':E \times E' \rightarrow X \times X'</math> with <math>(p \times p')(e, e') = (p(e), p'(e'))</math> is a covering.<ref name="Munkres">{{Cite book|last=Munkres|first=James|title=Topology|publisher=Upper Saddle River, NJ: Prentice Hall, Inc.|year=2000|isbn=978-0-13-468951-7}}</ref>{{rp|p=339}} However, coverings of <math>X\times X'</math> are not all of this form in general.

=== Equivalence of coverings ===
Let <math>X</math> be a topological space and <math>p:E \rightarrow X</math> and <math>p':E' \rightarrow X</math> be coverings. Both coverings are called '''equivalent''', if there exists a homeomorphism <math>h:E \rightarrow E'</math>, such that the diagram
[[File:Kommutatives_Diagramm_Äquivalenz_von_Überlagerungen.png|center|frameless]]
commutes. If such a homeomorphism exists, then one calls the covering spaces <math>E</math> and <math>E'</math> [[Isomorphism|isomorphic]].

=== Lifting property ===
All coverings satisfy the [[lifting property]], i.e.:

Let <math>I</math> be the [[unit interval]] and <math>p:E \rightarrow X</math> be a covering. Let <math>F:Y \times I \rightarrow X</math> be a continuous map and <math>\tilde F_0:Y \times \{0\} \rightarrow E</math> be a lift of <math>F|_{Y \times \{0\}}</math>, i.e. a continuous map such that <math>p \circ \tilde F_0 = F|_{Y \times \{0\}}</math>. Then there is a uniquely determined, continuous map <math>\tilde F:Y \times I \rightarrow E</math> for which <math>\tilde F(y,0) = \tilde F_0</math> and which is a lift of <math>F</math>, i.e. <math>p \circ \tilde F = F</math>.{{r|Hatcher|p=60}}

If <math>X</math> is a path-connected space, then for <math>Y=\{0\}</math> it follows that the map <math>\tilde F</math> is a lift of a [[Path (topology)|path]] in <math>X</math> and for <math>Y=I</math> it is a lift of a [[homotopy]] of paths in <math>X</math>.

As a consequence, one can show that the [[fundamental group]] <math>\pi_{1}(S^1)</math> of the unit circle is an [[Cyclic group|infinite cyclic group]], which is generated by the homotopy classes of the loop <math>\gamma: I \rightarrow S^1</math> with <math>\gamma (t) = (\cos(2 \pi t), \sin(2 \pi t))</math>.{{r|Hatcher|p=29}}

Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Let <math>x,y \in X</math> be any two points, which are connected by a path <math>\gamma</math>, i.e. <math>\gamma(0)= x</math> and <math>\gamma(1)= y</math>. Let <math>\tilde \gamma</math> be the unique lift of <math>\gamma</math>, then the map 
: <math>L_{\gamma}:p^{-1}(x) \rightarrow p^{-1}(y)</math> with <math>L_{\gamma}(\tilde \gamma (0))=\tilde \gamma (1)</math>

is [[Bijection|bijective]].{{r|Hatcher|p=69}}

If <math>X</math> is a path-connected space and <math>p: E \rightarrow X</math> a connected covering, then the induced [[group homomorphism]] 
: <math> p_{\#}: \pi_{1}(E) \rightarrow \pi_{1}(X)</math> with <math> p_{\#}([\gamma])=[p \circ \gamma]</math>,
is [[Injective function|injective]] and the [[subgroup]] <math>p_{\#}(\pi_1(E))</math> of <math>\pi_1(X)</math> consists of the homotopy classes of loops in <math>X</math>, whose lifts are loops in <math>E</math>.{{r|Hatcher|p=61}}