Editing Homeomorphism (section)

==Examples==
[[Image:Blue Trefoil Knot.png|thumb|upright=1.15|A thickened [[trefoil knot]] is homeomorphic to a solid torus, but not [[Homotopy#Isotopy|isotopic]] in  {{tmath|\R^3.}} Continuous mappings are not always realizable as deformations.]]
* The open [[interval (mathematics)|interval]] <math display="inline">(a,b)</math> is homeomorphic to the [[real number]]s {{tmath|\R}} for any <math display="inline">a < b.</math> (In this case, a bicontinuous forward mapping is given by <math display="inline">f(x) = \frac{1}{a-x} + \frac{1}{b-x} </math> while other such mappings are given by scaled and translated versions of the {{math|tan}} or {{math|arg tanh}} functions).
* The unit 2-[[ball (mathematics)|disc]] <math display="inline">D^2</math> and the [[unit square]] in {{tmath|\R^2}} are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in [[polar coordinates]], <math display=block>(\rho, \theta) \mapsto \left( \frac{\rho}{ \max(|\cos \theta|, |\sin \theta|)}, \theta\right).</math>
* The [[graph of a function|graph]] of a [[differentiable function]] is homeomorphic to the [[domain of a function|domain]] of the function.
* A differentiable [[parametric equation|parametrization]] of a [[curve]] is a homeomorphism between the domain of the parametrization and the curve.
* A [[chart (topology)|chart]] of a [[manifold]] is a homeomorphism between an [[open subset]] of the manifold and an open subset of a [[Euclidean space]].
* The [[stereographic projection]] is a homeomorphism between the unit sphere in {{tmath|\R^3}} with a single point removed and the set of all points in {{tmath|\R^2}} (a 2-dimensional [[plane (mathematics)|plane]]). 
* If <math>G</math> is a [[topological group]], its inversion map <math>x \mapsto x^{-1}</math> is a homeomorphism. Also, for any <math>x \in G,</math> the left translation <math>y \mapsto xy,</math> the right translation <math>y \mapsto yx,</math> and the inner automorphism <math>y \mapsto xyx^{-1}</math> are homeomorphisms.

===Counter-examples===
* {{tmath|\R^m}} and {{tmath|\R^n}} are not homeomorphic for {{math|1=''m'' &ne; ''n''.}}
* The Euclidean [[real line]] is not homeomorphic to the unit circle as a subspace of {{tmath|\R^2}}, since the unit circle is [[Compact space|compact]] as a subspace of Euclidean {{tmath|\R^2}} but the real line is not compact.
*The one-dimensional intervals <math>[0,1]</math> and <math>(0,1)</math> are not homeomorphic because one is compact while the other is not.