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PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION

Part of: Fairly general properties Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  03 July 2020

FLORICA C. CÎRSTEA Open the ORCID record for FLORICA C. CÎRSTEA [Opens in a new window]
FLORICA C. CÎRSTEA*
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Camperdown, NSW 2006, Australia email [email protected]
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Abstract

Urysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

Keywords

Urysohn’s lemmanormal spaceCantor set

MSC classification

Primary: 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
Secondary: 54C05: Continuous maps 54C99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 326 - 332
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Blair, R. L., ‘Proofs of Urysohn’s lemma and related theorems by means of Zorn’s lemma’, Math. Mag. 47(2) (1974), 71–78.Google Scholar
Cantor, G., ‘Über unendliche, lineare Punktmannichfaltigkeiten, Part 5’, Math. Ann. 21 (1883), 545–591.Google Scholar
Cantor, G., ‘De la puissance des ensembles parfaits de points’, Acta Math. 4 (1884), 381–392.Google Scholar
Dovgoshey, O., Martio, O., Ryazanov, V. and Vuorinen, M., ‘The Cantor function’, Expo. Math. 24 (2006), 1–37.Google Scholar
Fleron, J. F., ‘A note on the history of the Cantor set and Cantor function’, Math. Mag. 67 (1994), 136–140.Google Scholar
Folland, G. B., ‘A tale of topology’, Amer. Math. Monthly 117(8) (2010), 663–672.Google Scholar
Grabiner, S., ‘The Tietze extension theorem and the open mapping theorem’, Amer. Math. Monthly 93 (1986), 190–191.Google Scholar
Kelley, J. L., General Topology, Graduate Texts in Mathematics, 27 (Springer, New York, 1975), reprint of the 1955 edition, Van Nostrand, Toronto, ON.Google Scholar
Mandelbrot, B. B., The Fractal Geometry of Nature (W. H. Freeman, San Francisco, CA, 1982).Google Scholar
Mandelkern, M., ‘A short proof of the Tietze–Urysohn extension theorem’, Arch. Math. 60 (1993), 364–366.Google Scholar
Munkres, J. R., Topology, second edn (Prentice-Hall, Upper Saddle River, NJ, 2000).Google Scholar
Ossa, E., ‘A simple proof of the Tietze–Urysohn extension theorem’, Arch. Math. 71 (1998), 331–332.Google Scholar
Scott, B. M., ‘A “more topological” proof of the Tietze–Urysohn theorem’, Amer. Math. Monthly 85 (1978), 192–193.Google Scholar
Smith, H. J. S., ‘On the integration of discontinuous functions’, Proc. Lond. Math. Soc. (1) 6 (1875), 140–153.Google Scholar
Tietze, H., ‘Über Funktionen, die auf einer abgeschlossenen Menge stetig sind’, J. reine angew. Math. 145 (1915), 9–14.Google Scholar
Urysohn, P., ‘Über die Mächtigkeit der zusammenhängenden Mengen’, Math. Ann. 94 (1925), 262–295.Google Scholar
Urysohn, P., ‘Zum Metrisationsproblem’, Math. Ann. 94 (1925), 309–315.Google Scholar
Vallin, R. W., The Elements of Cantor Sets—With Applications (John Wiley, Hoboken, NJ, 2013).Google Scholar
Willard, S., General Topology (Dover, New York, 2004).Google Scholar