Editing Invariance of domain

{{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]].

==Notes==

The conclusion of the theorem can equivalently be formulated as: "<math>f</math> is an [[open map]]".

Normally, to check that <math>f</math> is a homeomorphism, one would have to verify that both <math>f</math> and its [[inverse function]] <math>f^{-1}</math> are continuous; 
the theorem says that if the domain is an {{em|open}} subset of <math>\R^n</math> and the image is also in <math>\R^n,</math> then continuity of <math>f^{-1}</math> is automatic. 
Furthermore, the theorem says that if two subsets <math>U</math> and <math>V</math> of <math>\R^n</math> are homeomorphic, and <math>U</math> is open, then <math>V</math> must be open as well. 
(Note that <math>V</math> is open as a subset of <math>\R^n,</math> and not just in the subspace topology. 
Openness of <math>V</math> in the subspace topology is automatic.) 
Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.

[[File:A map which is not a homeomorphism onto its image.png|thumb|alt=Not a homeomorphism onto its image|An injective map which is not a homeomorphism onto its image: <math>g : (-1.1, 1) \to \R^2</math> with <math>g(t) = \left(t^2 - 1, t^3 - t\right).</math>]]
It is of crucial importance that both [[Domain of a function|domain]] and [[Image of a function|image]] of <math>f</math> are contained in Euclidean space {{em|of the same dimension}}. 
Consider for instance the map <math>f : (0, 1) \to \R^2</math> defined by <math>f(t) = (t, 0).</math> 
This map is injective and continuous, the domain is an open subset of <math>\R</math>, but the image is not open in <math>\R^2.</math> 
A more extreme example is the map <math>g : (-1.1, 1) \to \R^2</math> defined by <math>g(t) = \left(t^2 - 1, t^3 - t\right)</math> because here <math>g</math> is injective and continuous but does not even yield a homeomorphism onto its image.

The theorem is also not generally true in infinitely many dimensions. Consider for instance the [[Banach space|Banach]] [[lp space|{{mvar|L<sup>p</sup>}} space]] <math>\ell^{\infty}</math> of all bounded real [[sequence]]s. 
Define <math>f : \ell^\infty \to \ell^\infty</math> as the shift <math>f\left(x_1, x_2, \ldots\right) = \left(0, x_1, x_2, \ldots\right).</math>  
Then <math>f</math> is injective and continuous, the domain is open in <math>\ell^{\infty}</math>, but the image is not.

==Consequences==

If <math>n>m</math>, there exists no continuous injective map <math>f:U\to\R^m</math> for a nonempty open set <math>U\subseteq\R^n</math>. To see this, suppose there exists such a map <math>f.</math>  Composing <math>f</math> with the standard inclusion of <math>\R^m</math> into <math>\R^n</math> would give a continuous injection from <math>\R^n</math> to itself, but with an image with empty interior in <math>\R^n</math>.  This would contradict invariance of domain.

In particular, if <math>n\ne m</math>, no nonempty open subset of <math>\R^n</math> can be homeomorphic to an open subset of <math>\R^m</math>.

And <math>\R^n</math> is not homeomorphic to <math>\R^m</math> if <math>n\ne m.</math>

==Generalizations==

The domain invariance theorem may be generalized to [[manifold]]s: if <math>M</math> and <math>N</math> are topological {{mvar|n}}-manifolds without boundary and <math>f : M \to N</math> is a continuous map which is [[Locally injective function|locally one-to-one]] (meaning that every point in <math>M</math> has a [[Neighborhood (topology)|neighborhood]] such that <math>f</math> restricted to this neighborhood is injective), then <math>f</math> is an [[open map]] (meaning that <math>f(U)</math> is open in <math>N</math> whenever <math>U</math> is an open subset of <math>M</math>) and a [[local homeomorphism]].

There are also generalizations to certain types of continuous maps from a [[Banach space]] to itself.<ref>{{aut|[[Jean Leray|Leray J.]]}} Topologie des espaces abstraits de M. Banach. ''C. R. Acad. Sci. Paris'', 200 (1935)  pages 1083–1093</ref>

==See also==

* [[Open mapping theorem]] for other conditions that ensure that a given continuous map is open.

==Notes==

{{reflist|30em}}

==References==

* {{cite book|mr=1224675|last=Bredon|first= Glen E. |title=Topology and geometry|series=Graduate Texts in Mathematics|volume= 139|publisher= [[Springer-Verlag]]|year= 1993|isbn=0-387-97926-3}}
* {{cite journal|mr=4101407|last=Cao Labora|first = Daniel | title= When is a continuous bijection a homeomorphism? |journal = [[Amer. Math. Monthly]]|volume= 127 |year=2020|issue= 6|pages= 547–553|
doi=10.1080/00029890.2020.1738826|s2cid=221066737 }}
* {{cite journal|mr=0013313|last= Cartan |first = Henri| title=Méthodes modernes en topologie algébrique| lang=fr| journal = [[Comment. Math. Helv.]]|volume= 18 
|year=1945|pages = 1–15|doi= 10.1007/BF02568096 |s2cid= 124671921 |url=https://link.springer.com/article/10.1007%2FBF02568096|url-access= subscription}}
* {{cite book|mr=3887626 | last=Deo |first= Satya | title = Algebraic topology: A primer|edition= Second |series = Texts and Readings in Mathematics| volume= 27| publisher =Hindustan Book Agency| location= New Delhi | year = 2018 | isbn = 978-93-86279-67-5}}
* {{cite book|mr=0658305|last=Dieudonné|first= Jean|title=Éléments d'analyse|volume= IX|series=Cahiers Scientifiques |publisher=Gauthier-Villars|location=Paris|year=1982|isbn= 2-04-011499-8|lang=fr|chapter=8. Les théorèmes de Brouwer|pages=44–47}}
* {{cite book |first=Morris W. |last=Hirsch | author-link=Morris Hirsch| title=Differential Topology |location=New York |publisher=Springer |year=1988 |isbn=978-0-387-90148-0 }} (see p.&nbsp;72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
* {{cite book|mr=0115161|last1=Hilton|first1= Peter J.|last2= Wylie|first2= Shaun|title=Homology theory: An introduction to algebraic topology|publisher=[[Cambridge University Press]]|location= New York |year=1960|isbn=0521094224}}
* {{cite book|mr=0006493|last1=Hurewicz|first1= Witold|last2= Wallman|first2= Henry|title=Dimension Theory|series=Princeton Mathematical Series|volume= 4|publisher=[[Princeton University Press]]| year=1941|url=https://archive.org/details/in.ernet.dli.2015.84609/}}
* {{cite journal|last=Kulpa|first=Władysław|title=Poincaré and domain invariance theorem|journal=Acta Univ. Carolin. Math. Phys.| volume=39|issue=1|year=1998|pages=129–136
|url=http://dml.cz/bitstream/handle/10338.dmlcz/702050/ActaCarolinae_039-1998-1_10.pdf|mr=1696596}}
* {{cite book|mr=1454127 |last1=Madsen|first1= Ib |last2= Tornehave|first2= Jørgen |title=From calculus to cohomology: de Rham cohomology and characteristic classes|publisher= [[Cambridge University Press]]|year= 1997|isbn= 0-521-58059-5}}
* {{cite book|mr=0198479 | last =Munkres| first = James R.|title=Elementary differential topology|series=Annals of Mathematics Studies|volume= 54 |publisher = [[Princeton University Press]]| year= 1966|edition=Revised}}
* {{cite book|last=Spanier|first= Edwin H.|title=Algebraic topology|publisher= McGraw-Hill |location= New York-Toronto-London|year= 1966}}
* {{cite web|url=http://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/|title=Brouwer's fixed point and invariance of domain theorems, and Hilbert's fifth problem|first=Terence|last=Tao|author-link=Terence Tao|work=terrytao.wordpress.com|year=2011|access-date=2 February 2022}}

==External links==

* {{SpringerEOM|title=Domain invariance|id=Domain_invariance|oldid=16623|first=J. van|last=Mill}}

{{Topology}}

[[Category:Algebraic topology]]
[[Category:Theory of continuous functions]]
[[Category:Homeomorphisms]]
[[Category:Theorems in topology]]