Editing Covering space (section)

=== Examples ===
[[File:Hawaiian_Earrings.svg|right|thumb|250x250px|The Hawaiian earring. Only the ten largest circles are shown.]]

* <math>r : \mathbb{R} \to S^1</math> with <math>r(t)=(\cos(2 \pi t), \sin(2 \pi t))</math> is the universal covering of the unit circle <math>S^1</math>.
* <math>p : S^n \to \mathbb{R}P^n \cong \{+1,-1\}\backslash S^n</math> with <math>p(x)=[x]</math> is the universal covering of the [[projective space]] <math>\mathbb{R}P^n</math> for <math>n>1</math>.
* <math>q : \mathrm{SU}(n) \ltimes \mathbb{R} \to U(n)</math> with <math display=block>q(A,t)= \begin{bmatrix}
 \exp(2 \pi i t) & 0\\
 0 & I_{n-1}
\end{bmatrix}_\vphantom{x} A </math> is the universal covering of the [[unitary group]] <math>U(n)</math>.<ref>{{Cite journal
|last1=Aguilar |first1=Marcelo Alberto
|last2=Socolovsky |first2=Miguel
|date=23 November 1999
|title=The Universal Covering Group of U(n) and Projective Representations
|journal=[[International Journal of Theoretical Physics]]
|publisher=Springer US
|publication-date=April 2000
|volume=39
|issue=4
|pages=997–1013
|arxiv=math-ph/9911028
|doi=10.1023/A:1003694206391
|bibcode=1999math.ph..11028A|s2cid=18686364
}}</ref>{{rp|p=5|at=Theorem 1}}
* Since <math>\mathrm{SU}(2) \cong S^3</math>, it follows that the [[quotient map (topology)|quotient map]] <math display=block>f : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2) /  \mathbb{Z_2} \cong \mathrm{SO}(3)</math> is the universal covering of <math>\mathrm{SO}(3)</math>.
* A topological space which has no universal covering is the [[Hawaiian earring]]: <math display=block>
X = \bigcup_{n\in \N}\left\{(x_1,x_2)\in\R^{2} : \Bigl(x_1-\frac{1}{n}\Bigr)^2+x_2^2=\frac{1}{n^2}\right\}
</math> One can show that no neighborhood of the origin <math>(0,0)</math> is simply connected.{{r|Munkres|p=487|at=Example 1}}