Proof of Converse of Urysohn's Lemma

Urysohn's Lemma

If a topological spaceis normal then, given any disjoint non empty closed subsetsandofthere is a continuous functionwherehas the absolute value topology, such that for everyand 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 ofThere is a continuous functionsuch that for all

The setsandare disjoint open sets subsets ofSinceis continuousandare disjoint open subsets ofsuch thatandhenceis normal.

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