Invariance of domain

Theorem in topology about homeomorphic subsets of Euclidean space

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space $\mathbb {R} ^{n}$ . It states:

U

is an open subset of

\mathbb {R} ^{n}

and

f:U\rightarrow \mathbb {R} ^{n}

is an injective continuous map, then

V:=f(U)

is open in

\mathbb {R} ^{n}

and

f

is a homeomorphism between

U

and

V

The theorem and its proof are due to L. E. J. Brouwer, published in 1912.^[1] The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

Notes

The conclusion of the theorem can equivalently be formulated as: " $f$ is an open map".

Normally, to check that $f$ is a homeomorphism, one would have to verify that both $f$ and its inverse function $f^{-1}$ are continuous; the theorem says that if the domain is an open subset of $\mathbb {R} ^{n}$ and the image is also in $\mathbb {R} ^{n},$ then continuity of $f^{-1}$ is automatic. Furthermore, the theorem says that if two subsets $U$ and $V$ of $\mathbb {R} ^{n}$ are homeomorphic, and $U$ is open, then $V$ must be open as well. (Note that $V$ is open as a subset of $\mathbb {R} ^{n},$ and not just in the subspace topology. Openness of $V$ 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.

Not a homeomorphism onto its image — A map which is not a homeomorphism onto its image: $g:(-1.1,1)\to \mathbb {R} ^{2}$ with $g(t)=\left(t^{2}-1,t^{3}-t\right).$

{\displaystyle g:(-1.1,1)\to \mathbb {R} ^{2}} — A map which is not a homeomorphism onto its image: $g:(-1.1,1)\to \mathbb {R} ^{2}$ with $g(t)=\left(t^{2}-1,t^{3}-t\right).$

It is of crucial importance that both domain and image of $f$ are contained in Euclidean space of the same dimension. Consider for instance the map $f:(0,1)\to \mathbb {R} ^{2}$ defined by $f(t)=(t,0).$ This map is injective and continuous, the domain is an open subset of $\mathbb {R}$ , but the image is not open in $\mathbb {R} ^{2}.$ A more extreme example is the map $g:(-1.1,1)\to \mathbb {R} ^{2}$ defined by $g(t)=\left(t^{2}-1,t^{3}-t\right)$ because here $g$ 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 L^p space $\ell ^{\infty }$ of all bounded real sequences. Define $f:\ell ^{\infty }\to \ell ^{\infty }$ as the shift $f\left(x_{1},x_{2},\ldots \right)=\left(0,x_{1},x_{2},\ldots \right).$ Then $f$ is injective and continuous, the domain is open in $\ell ^{\infty }$ , but the image is not.

Consequences

An important consequence of the domain invariance theorem is that $\mathbb {R} ^{n}$ cannot be homeomorphic to $\mathbb {R} ^{m}$ if $m\neq n.$ Indeed, no non-empty open subset of $\mathbb {R} ^{n}$ can be homeomorphic to any open subset of $\mathbb {R} ^{m}$ in this case.

Generalizations

The domain invariance theorem may be generalized to manifolds: if $M$ and $N$ are topological n-manifolds without boundary and $f:M\to N$ is a continuous map which is locally one-to-one (meaning that every point in $M$ has a neighborhood such that $f$ restricted to this neighborhood is injective), then $f$ is an open map (meaning that $f(U)$ is open in $N$ whenever $U$ is an open subset of $M$ ) and a local homeomorphism.

There are also generalizations to certain types of continuous maps from a Banach space to itself.^[2]

Notes

^ Brouwer L.E.J. Beweis der Invarianz des $n$ -dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305–315; see also 72 (1912), pages 55–56
^ Leray J. Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci. Paris, 200 (1935) pages 1083–1093

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