Tietze extension theorem

The Tietze extension theorem in topology states that, if X is a normal topological space and

f : A → 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 → 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.