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Homeomorphism
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{{Short description|Mapping which preserves all topological properties of a given space}} {{For|homeomorphisms in graph theory|Homeomorphism (graph theory)}} {{distinguish|Homomorphism}} {{redirect|Topological equivalence|the concept in dynamical systems|Topological conjugacy}} [[Image:Mug and Torus morph.gif|thumb|upright=1.2|An often-repeated [[mathematical joke]] is that topologists cannot tell the difference between a [[coffee mug]] and a [[donut]],<ref>{{cite book|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|first1=John H.|last1=Hubbard|first2=Beverly H.|last2=West|publisher=Springer|series=Texts in Applied Mathematics|volume=18|year=1995|isbn=978-0-387-94377-0|page=204|url=https://books.google.com/books?id=SHBj2oaSALoC&pg=PA204 }}</ref> since a sufficiently pliable [[donut]] could be reshaped to the form of a [[coffee mug]] by creating a dimple and progressively enlarging it, while preserving the donut hole in the mug's handle. This illustrates that a coffee mug and a donut ([[torus]]) are homeomorphic.]] In [[mathematics]] and more specifically in [[topology]], a '''homeomorphism''' ([[Neoclassical compound|from Greek roots]] meaning "similar shape", named by [[Henri Poincaré]]),<ref>{{cite book |url=http://serge.mehl.free.fr/anx/ana_situs.html |title=Analysis Situs |author-link= Henri Poincaré|last=Poincaré |first=H. |date=1895 |publisher=Gauthier-Villars |series=Journal de l'Ecole polytechnique |oclc=715734142 |access-date=29 April 2018|url-status=dead|archive-url=https://web.archive.org/web/20160611022329/http://serge.mehl.free.fr/anx/ana_situs.html|archive-date=11 June 2016}}<br/>{{cite book |last=Poincaré |first=Henri |year=2010 |title=Papers on Topology: Analysis Situs and Its Five Supplements |translator-first=John |translator-last=Stillwell |publisher=American Mathematical Society |isbn=978-0-8218-5234-7}}</ref><ref>{{cite book |last1=Gamelin |first1=T. W. |last2=Greene |first2=R. E. |year=1999 |title=Introduction to Topology |publisher=Dover |isbn=978-0-486-40680-0 |edition=2nd |page=67 |url=https://books.google.com/books?id=thAHAGyV2MQC&pg=PA67 }}</ref> also called '''topological isomorphism''', or '''bicontinuous function''', is a [[bijective]] and [[Continuous function#Continuous functions between topological spaces|continuous function]] between [[topological space]]s that has a continuous [[inverse function]]. Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]—that is, they are the [[map (mathematics)|mappings]] that preserve all the [[topological property|topological properties]] of a given space. Two spaces with a homeomorphism between them are called '''homeomorphic''', and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a [[geometry|geometric]] object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus]] are not. However, this description can be misleading. Some continuous deformations do not result into homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as the homeomorphism between a [[trefoil knot]] and a circle. [[Homotopy]] and [[homotopy#Isotopy|isotopy]] are precise definitions for the informal concept of ''continuous deformation''.
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