Editing Covering space (section)

=== 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}}