Invariance of domain

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

If

U

is an open subset of

ℝ n

and

f : U \to ℝ n

is an injective continuous map, then

V := f (U)

is open in

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 $ℝ n$ and the image is also in $ℝ n$ , then continuity of $f -1$ is automatic. Furthermore, the theorem says that if two subsets U and $V$ of $ℝ n$ are homeomorphic, and $U$ is open, then $V$ must be open as well. (Note that V is open as a subset of $ℝ n$ , 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.

A map which is not a homeomorphism onto its image:

g : (-1, 1) \to ℝ 2

with g(t) = (t² − 1, t³ − t)

It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → ℝ² defined by $f (t) = (t, 0)$ . This map is injective and continuous, the domain is an open subset of $ℝ$ , but the image is not open in $ℝ 2$ . A more extreme example is the map $g : (-1.1, 1) \to ℝ 2$ defined by $g (t) = (t 2 - 1, t 3 - t)$ 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 infinite dimensions. Consider for instance the Banach space l^∞ of all bounded real sequences. Define $f : l \infty \to l \infty$ as the shift $f (x 1, x 2, ...) = (0, x 1, x 2, ...)$ . Then $f$ is injective and continuous, the domain is open in $l \infty$ , but the image is not.

Consequences

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

References

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

External links

Mill, J. van (2001) [1994], "Domain invariance", Encyclopedia of Mathematics, EMS Press

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[1] 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

[2] Leray J. Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci. Paris, 200 (1935) pages 1083–1093