Domain invariance

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