A short proof of the Tietze-Urysohn extension theorem

Abstract

Tietze [8] proved the extension theorem for metric spaces, and Urysohn I10] for normal topological spaces. Urysohn first proves his Lemma, which is a special case of the theorem. The proof of the lemma uses a set-theoretic argument which constructs a family of sets indexed by the rationals, and defines a continuous real-valued function using infima of subsets of the indices. In rather surprising contrast, the full extension theorem then makes use of infinite series, the Weierstrass M-test, and uniform convergence. The purpose of this note is to extend the method of Urysohn's Lemma so as to obtain the extension theorem directly, without the use of uniform convergence, and without first proving the lemma. Urysohn's Lemma itself is then no longer required, being an immediate corollary of the theorem.