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.


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