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

If, there exists no continuous injective map for a nonempty open set. To see this, suppose there exists such a map Composing with the standard inclusion of into would give a continuous injection from to itself, but with an image with empty interior in. This would contradict invariance of domain.
In particular, if, no nonempty open subset of can be homeomorphic to an open subset of.
And is not homeomorphic to if

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.