# Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer 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. Moreover, F may be chosen such that $\sup \{ |f(a)| : a \in A \} = \sup \{ |F(x)| : x \in X \}$, 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.

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]

## References

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.