Proof of Condition for a Topological Space to be Normal

Theorem

A topological spaceis normal if and only if, for every closed setand every open setcontainingan open setexists such that

Proof

Letbe a normal space. Letbe a closed set andan open set inwith

is closed and

andare disjoint closed sets hence open setsandexist such thatand

Sincewe haveand sincewe have

The setis closed hence

Now letandbe disjoint closed sets thenandis open. An open set exists such that

Sincewe haveAlso sincewe have

Sinceis open,and whereand are open sets.