Invariance of domain

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

If U is an open subset of Rⁿ and f : U → Rⁿ is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and V.

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 Rⁿ and the image is also in Rⁿ, then continuity of f ^-1 is automatic. Furthermore, the theorem says that if two subsets U and V of Rⁿ are homeomorphic, and U is open, then V must be open as well. Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.

The theorem is due to L.E.J. Brouwer. Its proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

Consequences

An important consequence of the domain invariance theorem is that Rⁿ cannot be homeomorphic to R^m if m ≠ n. Indeed, no non-empty open subset of Rⁿ can be homeomorphic to any open subset of R^m in this case. (If n < m, then we can view Rⁿ as a subspace of R^m, and the non-empty open subsets of Rⁿ are not open as subsets of R^m.)

Generalization

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 M whenever U is an open subset of M).