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.