Invariance of domain
- 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.
The conclusion of the theorem can equivalently be formulated as: "f is an open map".
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. (Note that V is open as a subset of Rn, and not just in the subspace topology. Openness of V in the subspace topology is automatic. ) Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.
It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → R2 with f(t) = (t,0). This map is injective and continuous, the domain is an open subset of R, but the image is not open in R2. A more extreme example is g : (−1.1,1) → R2 with g(t) = (t 2 − 1, t 3 − t) because here g is injective and continuous but does not even yield a homeomorphism onto its image.
The theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l∞ of all bounded real sequences. Define f : l∞ → l∞ as the shift f(x1,x2,...) = (0, x1, x2,...). Then f is injective and continuous, the domain is open in l∞, but the image is not.
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.
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 N whenever U is an open subset of M) and a local homeomorphism.
- Open mapping theorem for other conditions that ensure that a given continuous map is open.