Theorem
Let
be a T4 space and let
be any closed subset of
If
is any continuous function fromto the set of real numbers with the absolute value topology then there is a continuous extension
of
from
to
Proof
Letbe a T4 space and let
and
be disjoint closed, nonempty subsets ofDefine
and
by
for
andfor
is continuous so according to Tietze's theorem, it can be extended to
resulting in a functionon