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 $$f : U \rarr \R^n$$ 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. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

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.