Editing Homeomorphism (section)

==Definition==
A [[function (mathematics)|function]] <math>f : X \to Y</math> between two [[topological space]]s is a '''homeomorphism''' if it has the following properties:

* <math>f</math> is a [[bijection]] ([[injective function|one-to-one]] and [[onto]]),
* <math>f</math> is [[Continuity (topology)|continuous]],
* the [[inverse function]] <math>f^{-1}</math> is continuous (<math>f</math> is an [[open mapping]]).

A homeomorphism is sometimes called a ''bicontinuous'' function. If such a function exists, <math>X</math> and <math>Y</math> are '''homeomorphic'''. A '''self-homeomorphism''' is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an [[equivalence relation]] on topological spaces. Its [[equivalence class]]es are called '''homeomorphism classes'''.

The third requirement, that <math display="inline">f^{-1}</math> be [[Continuous function|continuous]], is essential. Consider for instance the function <math display="inline">f : [0,2\pi) \to S^1</math> (the [[unit circle]] in {{tmath|\R^2}}) defined by<math display="inline">f(\varphi) = (\cos\varphi,\sin\varphi).</math> This function is bijective and continuous, but not a homeomorphism (<math display="inline">S^1</math> is [[Compact space|compact]] but <math display="inline">[0,2\pi)</math> is not). The function <math display="inline">f^{-1}</math> is not continuous at the point <math display="inline">(1,0),</math> because although <math display="inline">f^{-1}</math> maps <math display="inline">(1,0)</math> to <math display="inline">0,</math> any [[Neighbourhood (mathematics)|neighbourhood]] of this point also includes points that the function maps close to <math display="inline">2\pi,</math> but the points it maps to numbers in between lie outside the neighbourhood.<ref>{{cite book |last=Väisälä |first=Jussi |title=Topologia I |publisher=Limes RY |date=1999 |page=63  |isbn=951-745-184-9}}</ref>

Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms <math display="inline">X \to X</math> forms a [[group (mathematics)|group]], called the '''[[homeomorphism group]]''' of ''X'', often denoted <math display="inline">\operatorname{Homeo}(X).</math> This group can be given a topology, such as the [[compact-open topology]], which under certain assumptions makes it a [[topological group]].<ref>{{cite journal|last1=Dijkstra|first1=Jan J.|title=On Homeomorphism Groups and the Compact-Open Topology|journal=The American Mathematical Monthly|date=1 December 2005|volume=112|issue=10|pages=910–912|doi=10.2307/30037630|jstor=30037630 |url=http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf|url-status=live|archive-url=https://web.archive.org/web/20160916112245/http://www.cs.vu.nl/~dijkstra/research/papers/2005compactopen.pdf|archive-date=16 September 2016}}</ref>

In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other. [[Homotopy]] and  [[Homotopy#Isotopy|isotopy]] are equivalence relations that have been introduced for dealing with such situations.

Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, <math display="inline">\operatorname{Homeo}(X,Y),</math> is a [[torsor]] for the homeomorphism groups <math display="inline">\operatorname{Homeo}(X)</math> and <math display="inline">\operatorname{Homeo}(Y),</math> and, given a specific homeomorphism between <math>X</math> and <math>Y,</math> all three sets are identified.{{clarify|reason=Which?|date=July 2023}}