Urysohn's Lemma
If a topological space
is normal then, given any disjoint non empty closed subsets
and
of
there is a continuous function
where
has the absolute value topology, such that for every
and for every
Proof That the Converse of Urysohn's Lemma is true
Supposehas the property described in Urysohn's Lemma. Letandbe non empty closed subsets of
There is a continuous function
such that for all
The sets
and
are disjoint open sets subsets of
Since
is continuous
and
are disjoint open subsets of
such that
and
henceis normal.