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A short proof of the Tietze-Urysohn extension theorem. (English) Zbl 0778.54005
Tietze proved the extension theorem for metric spaces, and Urysohn 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. This paper extends 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.

54C30 Real-valued functions in general topology
54C20 Extension of maps
Full Text: DOI
[1] R. L. Blair, Proofs of Urysohn’s Lemma and related theorems by means of Zorn’s Lemma. Math. Mag.47, 71-78 (1974). · Zbl 0277.54017
[2] R. L. Blair andA. W. Hager, Extensions of zero-sets and of real-valued functions. Math. Z.136, 41-52 (1974). · Zbl 0271.54008
[3] E. Bonan, Sur un lemme adapté au théorème de Tietze-Urysohn. C. R. Acad. Sci. Paris Sér. A-B270, 1226-1228 (1970). · Zbl 0193.51103
[4] R.Engelking, General Topology. Warszawa 1977.
[5] S. Grabiner, The Tietze Extension Theorem and the open mapping theorem. Amer. Math. Monthly93, 190-191 (1986). · Zbl 0637.46004
[6] M. Kat?tov, On real-valued functions in topological spaces. Fund. Math.38, 85-91 (1951); Corrections in: Fund. Math.40, 203-205 (1953). · Zbl 0045.25704
[7] B. M. Scott, A ?more topological? proof of the Tietze-Urysohn Theorem. Amer. Math. Monthly85, 192-193 (1978). · Zbl 0376.54006
[8] H. Tietze, Über Funktionen, die auf einer abgeschlossenen Menge stetig sind. J. Reine Angew. Math.145, 9-14 (1915). · JFM 45.0319.01
[9] H. Tong, Some characterizations of normal and perfectly normal spaces. Duke Math. J.19, 289-292 (1952). · Zbl 0046.16203
[10] P. Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen. Math. Ann.94, 262-295 (1925). · JFM 51.0452.05
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