Statement and Proof of the Tietze Extension Theorem

PlanetMath: proof of Tietze extension theorem
by for and for
. Now is continuous and we can extend it to a continuous
function . By Urysohn's lemma, is normal because
is a continuous function such that for and
for .
Conversely, let be normal and be closed in . By the lemma,
there is a continuous function such that for
and for . Since
is continuous, the lemma tells us there is a
continuous function such that for
and for . By
repeated application of the lemma we can construct a sequence of
continuous functions such that for all
.
, and for
Define . Since and
converges as a geometric series, then
converges absolutely and uniformly, so is a continuous function
defined everywhere. Moreover implies that
.
Page 2 of 4
Now for , we have that and
as goes to infinity, the right side goes to zero and so the sum goes
to . Thus Therefore extends .
http://planetmath.org/encyclopedia/ProofOfTietzeExtensionTheorem2.html
9/26/2006