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 R^J for some indexing set J, any retract of R^J, or any normal absolute retract whatsoever.

The theorem is due to Heinrich Franz Friedrich Tietze.

References

• Weisstein, Eric W. "Tietze's Extension Theorem." From MathWorld
• Tietze extension theorem on PlanetMath
• Proof of Tietze extension theorem on PlanetMath