Invariance of domain

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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

If U is an open subset of and is an injective continuous map, then V := f(U) is open in 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[edit]

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 and the image is also in then continuity of f −1 is automatic. Furthermore, the theorem says that if two subsets U and V of are homeomorphic, and U is open, then V must be open as well. (Note that V is open as a subset of 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.

Not a homeomorphism onto its image
g : (−1, 1) → with g(t) = (t2 − 1, t3t)

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) defined by f(t) = (t, 0). This map is injective and continuous, the domain is an open subset of , but the image is not open in A more extreme example is the map g: (−1.1, 1) → defined by 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 : ll 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[edit]

An important consequence of the domain invariance theorem is that cannot be homeomorphic to if mn. Indeed, no non-empty open subset of can be homeomorphic to any open subset of in this case.

Generalizations[edit]

The domain invariance theorem may be generalized to manifolds: if M and N are topological n-manifolds without boundary and f : MN 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.

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

See also[edit]

References[edit]

  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[edit]

  • Mill, J. van (2001) [1994], "Domain invariance", Encyclopedia of Mathematics, EMS Press