Tietze extension theorem


Tietze extension theorem

In topology, the Tietze extension theorem states that, if X is a normal topological space and

f: A \to \mathbb{R}

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

F: X \to \mathbb{R}

with F(a) = f(a) for all a in A. F is called a continuous extension of f.

The theorem generalizes Urysohn's lemma 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.

The theorem is due to Heinrich Franz Friedrich Tietze.

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Tietze — Tietze: Tietze extension theorem Tietze s graph Tietze s syndrome, named after Alexander Tietze Tietze transformation in mathematics, named after Heinrich Tietze Tietze, Tieze as German surnames This disambiguation page lists articles associated… …   Wikipedia

  • Extension (mathematics) — In mathematics, the word extension has many uses. See:Analysis* Carathéodory s extension theorem * Continuous linear extension * M. Riesz extension theorem * Krein extension theorem * Hahn Banach theoremAlgebra* Abelian extension * Algebraic… …   Wikipedia

  • Heinrich Franz Friedrich Tietze — (August 31, 1880 ndash; February 17, 1964) was an Austrian mathematician. He was born in Schleinz, Austria and died in Munich, Germany.He is famous for the so called Tietze extension theorem.External links*MacTutor Biography|id=Tietze… …   Wikipedia

  • Katětov–Tong insertion theorem — The Katětov–Tong insertion theorem is a theorem of point set topology proved independently by Miroslav Katětov [Miroslav Katětov, On real valued functions in topological spaces , Fundamenta Mathematicae 38 (1951), 85–91.… …   Wikipedia

  • Théorème de prolongement de Tietze — Le mathématicien Pavel Urysohn généralise le résultat de Heinrich Tietze aux espace normaux. En mathématiques, le théorème de prolongement de Tietze encore appelé de Tietze Urysohn est un résultat de topologie. Ce théorème indique qu une fonction …   Wikipédia en Français

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie …   Wikipedia

  • Uniform continuity — In mathematical analysis, a function f ( x ) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f ( x ) ( continuity ), and furthermore the size of the changes in f ( x ) depends… …   Wikipedia

  • Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia

  • Space-filling curve — 3 iterations of a Peano curve construction, whose limit is a space filling curve. In mathematical analysis, a space filling curve is a curve whose range contains the entire 2 dimensional unit square (or more generally an N dimensional hypercube) …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.