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.
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