Invariance of domain

Template:Short description Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space <math>\R^n</math>. It states:

If <math>U</math> is an 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>Template:Aut 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 Template:Em 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

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 and image of <math>f</math> are contained in Euclidean space Template:Em. 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 [[lp space|Template:Mvar space]] <math>\ell^{\infty}</math> of all bounded real sequences. 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 manifolds: if <math>M</math> and <math>N</math> are topological Template:Mvar-manifolds without boundary and <math>f : M \to N</math> is a continuous map which is locally one-to-one (meaning that every point in <math>M</math> has a 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>Template:Aut 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.

References

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External links

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