Editing Invariance of domain (section)

{{Short description|Theorem in topology about homeomorphic subsets of Euclidean space}}
'''Invariance of domain''' is a theorem in [[topology]] about [[homeomorphic]] [[subset]]s of [[Euclidean space]] <math>\R^n</math>. 
It states: 
:If <math>U</math> is an [[Open set|open subset]] of <math>\R^n</math> and <math>f : U \rarr \R^n</math> is an [[injective]] [[continuous map]], then <math>V := f(U)</math> is open in <math>\R^n</math> and <math>f</math> is a [[homeomorphism]] between <math>U</math> and <math>V</math>.

The theorem and its proof are due to [[L. E. J. Brouwer]], published in 1912.<ref>{{aut|[[L.E.J. Brouwer|Brouwer L.E.J.]]}} Beweis der Invarianz des <math>n</math>-dimensionalen Gebiets, ''[[Mathematische Annalen]]'' 71 (1912), pages 305–315; see also 72 (1912), pages 55–56</ref> 
The proof uses tools of [[algebraic topology]], notably the [[Brouwer fixed point theorem]].