Tietze extension theorem
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.
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.
The theorem is due to Heinrich Franz Friedrich Tietze.
- Hazewinkel, Michiel, ed. (2001), "Urysohn-Brouwer lemma", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W. "Tietze's Extension Theorem." From MathWorld
- Tietze extension theorem at PlanetMath.org.
- Proof of Tietze extension theorem at PlanetMath.org.
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