Tietze extension theorem

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In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

Formal statement[edit]

If X is a normal topological space and

is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map

with F(a) = f(a) for all a in A. Moreover, F may be chosen such that , i.e., if f is bounded, F may be chosen to be bounded (with the same bound as f). F is called a continuous extension of f.

History[edit]

L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when X is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Paul Urysohn proved the theorem as stated here, for normal topological spaces.[1][2]

Equivalent statements[edit]

This theorem is equivalent to the Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing R with RJ for some indexing set J, any retract of RJ, or any normal absolute retract whatsoever.

Variations[edit]

If X is a metric space, A a non-empty subset of X and is a Lipschitz continuous function with Lipschitz constant K, then f can be extended to a Lipschitz continuous function with same constant K. This theorem is also valid for Hölder continuous functions, that is, if is Hölder continuous function, f can be extended to a Hölder continuous function with the same constant. [3]

Another variant (in fact, generalization) of Tietze's theorem is due to Z. Ercan:[4] Let A be a closed subset of a topological space X. If is an upper semicontinuous function, a lower semicontinuous function and a continuous function such that f(x)≤g(x) for each xX and f(a)≤h(a)≤g(a) for each aA, then there is a continuous extension of h such that f(x)≤H(x)≤g(x) for each xX. This theorem is also valid with some additional hypothesis if R is replaced by a general locally solid Riesz space.[4]

See also[edit]

References[edit]

  1. ^ Hazewinkel, Michiel, ed. (2001), "Urysohn-Brouwer lemma", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  2. ^ Urysohn, Paul (1925), "Über die Mächtigkeit der zusammenhängenden Mengen", Mathematische Annalen 94 (1): 262–295, doi:10.1007/BF01208659 .
  3. ^ McShane, E. J. (1 December 1934). "Extension of range of functions". Bulletin of the American Mathematical Society 40 (12): 837–843. doi:10.1090/S0002-9904-1934-05978-0. 
  4. ^ a b Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics 21 (4): 423–430. 

External links[edit]