# 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*→**R**^{n}is an injective continuous map, then*V*=*f*(*U*) is open and*f*is a homeomorphism between*U*and*V*.

The theorem and its proof are due to L. E. J. Brouwer, published in 1912.^{} 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.

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) → **R**^{2} with *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 *g* : (−1.1,1) → **R**^{2} with *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*^{∞} → *l*^{∞} 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*^{∞}, 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* ≠ *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* → *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.^{}

## See also

- Open mapping theorem for other conditions that ensure that a given continuous map is open.

## References

## External links

- Mill, J. van (2001), "Domain invariance", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4