The theorem is due to L.E.J. Brouwer. Its proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.
An important consequence of the domain invariance theorem is that Rn cannot be homeomorphic to Rm if m ≠ n. Indeed, no non-empty open subset of Rn can be homeomorphic to any open subset of Rm in this case. (If n < m, then we can view Rn as a subspace of Rm, and the non-empty open subsets of Rn are not open as subsets of Rm.)
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).Consequences
Generalization