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 R^{n} cannot be homeomorphic to R^{m} if m ≠ n. Indeed, no nonempty open subset of R^{n} can be homeomorphic to any open subset of R^{m} in this case. (If n < m, then we can view R^{n} as a subspace of R^{m}, and the nonempty open subsets of R^{n} are not open as subsets of R^{m}.)
The domain invariance theorem may be generalized to manifolds: if M and N are topological nmanifolds without boundary and f : M → N is a continuous map which is locally onetoone (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).
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