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.

The theorem and its proof are due to L.E.J. Brouwer, published in 1912.^[1] The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

Notes[link]

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) = (t²-1, t³-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.

Consequences[link]

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. (Proof: If m < n, then we can view R^m as a subspace of Rⁿ, and the non-empty open subsets of R^m are not open when considered as subsets of Rⁿ. We apply the theorem in the space Rⁿ.)...

Generalizations[link]

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).

There are also generalizations to certain types of continuous maps from a Banach space to itself.^[2]

See also[link]

• Open mapping theorem for other conditions that ensure that a given continuous map is open.

References[link]

1. ^ Brouwer L.E.J. Beweis der Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305-315; see also 72 (1912), pages 55-56
2. ^ Leray J. Topologie des espaces abstraits de M. Banach. C.R. Acad. Sci. Paris, 200 (1935) pages 1083–1093

External links[link]

• Mill, J. van (2001), "Domain invariance", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Domain_invariance