Invariance of domain


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

Consequences

An important consequence of the domain invariance theorem is that cannot be homeomorphic to if.
Indeed, no non-empty open subset of can be homeomorphic to any open subset of in this case.

Generalizations

The domain invariance theorem may be generalized to manifolds: if and are topological -manifolds without boundary and is a continuous map which is locally one-to-one, then is an open map and a local homeomorphism.
There are also generalizations to certain types of continuous maps from a Banach space to itself.