The Tietze extension theorem reference article from the English Wikipedia on 24-Jul-2004
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Tietze extension theorem

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The Tietze extension theorem in topology states that, if X is a normal topological space and
f : AR
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 : XR
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.