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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====
<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>