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Proof That Urysohn's Lemma is a Special Case of Tietze's Extension Theorem

Theorem

Letbe a T4 space and letbe any closed subset ofIfis any continuous function fromto the set of real numbers with the absolute value topology then there is a continuous extensionoffromto

Proof

Letbe a T4 space and letandbe disjoint closed, nonempty subsets ofDefine

and

byforandfor

is continuous so according to Tietze's theorem, it can be extended toresulting in a functionon