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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202501.png" /> be open. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202502.png" /> is a one-to-one [[Continuous function|continuous function]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202503.png" /> is open and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202504.png" /> is a [[Homeomorphism|homeomorphism]]. This is called the Brouwer invariance of domain theorem, and was proved by L.E.J. Brouwer in [[#References|[a1]]]. This result immediately implies that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202505.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202507.png" /> are not homeomorphic. A similar result for infinite-dimensional vector spaces does not hold, as the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d1202509.png" /> shows. But Brouwer's theorem can be extended to compact fields in Banach spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d12025011.png" />, as was shown by J. Schauder [[#References|[a3]]]. Here, a compact field (a name coming from  "compact vector field" ) is a mapping of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d12025012.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d12025013.png" /> a compact mapping. A more general result for arbitrary Banach spaces was established (by using degree theory for compact fields) by J. Leray [[#References|[a2]]]. Several important results in the theory of differential equations were proved by using domain invariance as a tool.
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Let $U \subseteq \mathbf{R} ^ { n }$ be open. If $f : U \rightarrow {\bf R} ^ { n }$ is a one-to-one [[Continuous function|continuous function]], then $f [ U ]$ is open and $f : U \rightarrow f [ U ]$ is a [[Homeomorphism|homeomorphism]]. This is called the Brouwer invariance of domain theorem, and was proved by L.E.J. Brouwer in [[#References|[a1]]]. This result immediately implies that if $n \neq m$, then ${\bf R} ^ { n }$ and $\mathbf{R} ^ { m }$ are not homeomorphic. A similar result for infinite-dimensional vector spaces does not hold, as the subspace $\{ x \in {\bf l} ^ { 2 } : x _ { 1 } = 0 \}$ of $\text{l} ^ { 2 }$ shows. But Brouwer's theorem can be extended to compact fields in Banach spaces of type $( S )$, as was shown by J. Schauder [[#References|[a3]]]. Here, a compact field (a name coming from  "compact vector field" ) is a mapping of the form $x \rightarrow x - \phi ( x )$, with $\phi$ a compact mapping. A more general result for arbitrary Banach spaces was established (by using degree theory for compact fields) by J. Leray [[#References|[a2]]]. Several important results in the theory of differential equations were proved by using domain invariance as a tool.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Invariantz des <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120250/d12025014.png" />-dimensionalen Gebiets"  ''Math. Ann.'' , '''71/2'''  (1912/3)  pp. 305–313; 55–56</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Leray,  "Topologie des espaces abstraits de M. Banach"  ''C.R. Acad. Sci. Paris'' , '''200'''  (1935)  pp. 1083–1093</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Schauder,  "Invarianz des Gebietes in Funktionalräumen"  ''Studia Math.'' , '''1'''  (1929)  pp. 123–139</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  L.E.J. Brouwer,  "Invariantz des $n$-dimensionalen Gebiets"  ''Math. Ann.'' , '''71/2'''  (1912/3)  pp. 305–313; 55–56</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Leray,  "Topologie des espaces abstraits de M. Banach"  ''C.R. Acad. Sci. Paris'' , '''200'''  (1935)  pp. 1083–1093</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Schauder,  "Invarianz des Gebietes in Funktionalräumen"  ''Studia Math.'' , '''1'''  (1929)  pp. 123–139</td></tr></table>

Latest revision as of 16:45, 1 July 2020

Let $U \subseteq \mathbf{R} ^ { n }$ be open. If $f : U \rightarrow {\bf R} ^ { n }$ is a one-to-one continuous function, then $f [ U ]$ is open and $f : U \rightarrow f [ U ]$ is a homeomorphism. This is called the Brouwer invariance of domain theorem, and was proved by L.E.J. Brouwer in [a1]. This result immediately implies that if $n \neq m$, then ${\bf R} ^ { n }$ and $\mathbf{R} ^ { m }$ are not homeomorphic. A similar result for infinite-dimensional vector spaces does not hold, as the subspace $\{ x \in {\bf l} ^ { 2 } : x _ { 1 } = 0 \}$ of $\text{l} ^ { 2 }$ shows. But Brouwer's theorem can be extended to compact fields in Banach spaces of type $( S )$, as was shown by J. Schauder [a3]. Here, a compact field (a name coming from "compact vector field" ) is a mapping of the form $x \rightarrow x - \phi ( x )$, with $\phi$ a compact mapping. A more general result for arbitrary Banach spaces was established (by using degree theory for compact fields) by J. Leray [a2]. Several important results in the theory of differential equations were proved by using domain invariance as a tool.

References

[a1] L.E.J. Brouwer, "Invariantz des $n$-dimensionalen Gebiets" Math. Ann. , 71/2 (1912/3) pp. 305–313; 55–56
[a2] J. Leray, "Topologie des espaces abstraits de M. Banach" C.R. Acad. Sci. Paris , 200 (1935) pp. 1083–1093
[a3] J. Schauder, "Invarianz des Gebietes in Funktionalräumen" Studia Math. , 1 (1929) pp. 123–139
How to Cite This Entry:
Domain invariance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Domain_invariance&oldid=49968
This article was adapted from an original article by J. van Mill (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article