Invariance of domain

(Redirected from Domain invariance theorem)

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

If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and .

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

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The conclusion of the theorem can equivalently be formulated as: "  is an open map".

Normally, to check that   is a homeomorphism, one would have to verify that both   and its inverse function   are continuous; the theorem says that if the domain is an open subset of   and the image is also in   then continuity of   is automatic. Furthermore, the theorem says that if two subsets   and   of   are homeomorphic, and   is open, then   must be open as well. (Note that   is open as a subset of   and not just in the subspace topology. Openness of   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.

 
A map which is not a homeomorphism onto its image:   with  

It is of crucial importance that both domain and image of   are contained in Euclidean space of the same dimension. Consider for instance the map   defined by   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   defined by   because here   is injective and continuous but does not even yield a homeomorphism onto its image.

The theorem is also not generally true in infinitely many dimensions. Consider for instance the Banach Lp space   of all bounded real sequences. Define   as the shift   Then   is injective and continuous, the domain is open in  , but the image is not.

Consequences

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

Generalizations

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The domain invariance theorem may be generalized to manifolds: if   and   are topological n-manifolds without boundary and   is a continuous map which is locally one-to-one (meaning that every point in   has a neighborhood such that   restricted to this neighborhood is injective), then   is an open map (meaning that   is open in   whenever   is an open subset of  ) and a local homeomorphism.

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

See also

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Notes

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  1. ^ Brouwer L.E.J. Beweis der Invarianz des  -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

References

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  • Bredon, Glen E. (1993). Topology and geometry. Graduate Texts in Mathematics. Vol. 139. Springer-Verlag. ISBN 0-387-97926-3. MR 1224675.
  • Cao Labora, Daniel (2020). "When is a continuous bijection a homeomorphism?". Amer. Math. Monthly. 127 (6): 547–553. doi:10.1080/00029890.2020.1738826. MR 4101407. S2CID 221066737.
  • Cartan, Henri (1945). "Méthodes modernes en topologie algébrique". Comment. Math. Helv. (in French). 18: 1–15. doi:10.1007/BF02568096. MR 0013313. S2CID 124671921.
  • Deo, Satya (2018). Algebraic topology: A primer. Texts and Readings in Mathematics. Vol. 27 (Second ed.). New Delhi: Hindustan Book Agency. ISBN 978-93-86279-67-5. MR 3887626.
  • Dieudonné, Jean (1982). "8. Les théorèmes de Brouwer". Éléments d'analyse. Cahiers Scientifiques (in French). Vol. IX. Paris: Gauthier-Villars. pp. 44–47. ISBN 2-04-011499-8. MR 0658305.
  • Hirsch, Morris W. (1988). Differential Topology. New York: Springer. ISBN 978-0-387-90148-0. (see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
  • Hilton, Peter J.; Wylie, Shaun (1960). Homology theory: An introduction to algebraic topology. New York: Cambridge University Press. ISBN 0521094224. MR 0115161.
  • Hurewicz, Witold; Wallman, Henry (1941). Dimension Theory. Princeton Mathematical Series. Vol. 4. Princeton University Press. MR 0006493.
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  • Madsen, Ib; Tornehave, Jørgen (1997). From calculus to cohomology: de Rham cohomology and characteristic classes. Cambridge University Press. ISBN 0-521-58059-5. MR 1454127.
  • Munkres, James R. (1966). Elementary differential topology. Annals of Mathematics Studies. Vol. 54 (Revised ed.). Princeton University Press. MR 0198479.
  • Spanier, Edwin H. (1966). Algebraic topology. New York-Toronto-London: McGraw-Hill.
  • Tao, Terence (2011). "Brouwer's fixed point and invariance of domain theorems, and Hilbert's fifth problem". terrytao.wordpress.com. Retrieved 2 February 2022.
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