Editing Covering space (section)

== Examples ==
* For every topological space <math>X</math>, the [[Identity function|identity map]] <math>\operatorname{id}:X \rightarrow X</math> is a covering. Likewise for any discrete space <math>D</math> the projection <math>\pi:X \times D \rightarrow X</math> taking <math>(x, i) \mapsto x</math> is a covering. Coverings of this type are called '''trivial coverings'''; if <math>D</math> has finitely many (say <math>k</math>) elements, the covering is called the '''trivial ''<math>k</math>-sheeted'' covering''' of <math>X</math>. 
{{Dark mode invert|[[File:Covering_map.svg|thumb|The space <math>Y=[0,1] \times \mathbb{R}</math> is a covering space of <math>X=[0,1] \times S^1</math>. The disjoint open sets <math>S_i</math> are mapped homeomorphically onto <math>U</math>. The fiber of <math>x</math> consists of the points <math>y_i</math>.]]}}
* The map <math>r : \mathbb{R} \to S^1</math> with <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math> is a covering of the [[unit circle]] <math>S^1</math>. The base of the covering is <math>S^1</math> and the covering space is <math>\mathbb{R}</math>. For any point <math>x = (x_1, x_2) \in S^1</math> such that <math>x_1 > 0</math>, the set <math>U := \{(x_1, x_2) \in S^1 \mid x_1 > 0 \}</math> is an open neighborhood of <math>x</math>. The preimage of <math>U</math> under <math>r</math> is 
*: <math>r^{-1}(U)=\displaystyle\bigsqcup_{n \in \mathbb{Z}} \left( n - \frac 1 4, n + \frac 1 4\right)</math> 
: and the sheets of the covering are <math>V_n = (n - 1/4, n+1/4)</math> for <math>n \in \mathbb{Z}.</math> The fiber of <math>x</math> is
:: <math>r^{-1}(x) = \{t \in \mathbb{R} \mid (\cos(2 \pi t), \sin(2 \pi t)) = x\}.</math> 
* Another covering of the unit circle is the map <math>q : S^1 \to S^1</math> with <math>q(z)=z^{n}</math> for some positive <math>n \in \mathbb{N}.</math> For an open neighborhood <math>U</math> of an <math>x \in S^1</math>, one has: 
:: <math>q^{-1}(U)=\displaystyle\bigsqcup_{i=1}^{n} U</math>.
* A map which is a [[local homeomorphism]] but not a covering of the unit circle is <math>p : \mathbb{R_{+}} \to S^1</math> with <math>p(t)=(\cos(2 \pi t), \sin(2 \pi t))</math>. There is a sheet of an open neighborhood of <math>(1,0)</math>, which is not mapped homeomorphically onto <math>U</math>.