A fundamental extension theorem of McShane states that a bounded real-valued uniformly continuous function defined on a nonempty subset A of a metric space 〈X, d〉 can be extended to a uniformly continuous function on the entire space. In the first half of this note, we obtain McShane’s Extension Theorem from the simpler fact that a real-valued Lipschitz function defined on a nonempty subset of the space has a Lipschitz constant preserving extension to the entire space. In the second half of the note, we use this theorem to give an elementary proof of the equivalence of the most important characterizations of metric spaces in which the real-valued uniformly continuous functions form a ring. These characterizations of such a basic property, due to Cabello-Sánchez and separately Bouziad and Sukhacheva, are remarkably recent.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Atsuji, M.: Uniform continuity of continuous functions of metric spaces. Pac. J. Math 8, 11–16 (1958)
Beer, G., Ceniceros, J.: Lipschitz functions and Ekeland’s theorem. J. Optim. Theory Appl. 152, 652–660 (2012)
Beer, G., Garrido, M.I.: Bornologies and locally Lipschitz functions. Bull. Aust. Math. Soc. 90, 257–263 (2014)
Beer, G., Garrido, M.I., Meroño, A.S.: Uniform continuity and a new bornology for a metric space. Set-valued Var. Anal. 26, 49–65 (2018)
Beer, G., Levi, S.: Strong uniform continuity. J. Math. Anal. Appl. 350, 568–589 (2009)
Bourbaki, N.: Topologie Générale. Hermann, Paris (1965)
Bouziad, A., Sukhacheva, E.: Preservation of uniform continuity under pointwise product. Topol. Appl. 254, 132–144 (2019)
Cabello-Sánchez, J.: U(X) as a ring for metric spaces X. Filomat 31, 1981–1984 (2017)
Corazza, P.: Introduction to metric-preserving functions. Am. Math. Mon. 106, 309–323 (1999)
Dugundji, J.: Topology. Allyn and Bacon, Boston (1966)
Garrido, M.I., Meroño, A.S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39, 733–758 (2014)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Hejcman, J.: Boundedness in uniform spaces and topological groups. Czech. Math. J. 9, 544–563 (1959)
Hogbe-Nlend, H.: Bornologies and Functional Analysis. North-Holland, Amsterdam (1977)
Katetov, M.: On real-valued functions in topological spaces. Fund. Math. 38, 85–91 (1951)
Kelley, J.L.: General Topology. Van Nostrand, Princeton (1955)
McShane, E.J.: Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)
Roberts, A.W., Varberg, D.: Convex Functions. Academic Press, New York (1973)
Sullivan, F.: A characterization of complete metric spaces. Proc. Am. Math. Soc. 83, 345–346 (1981)
Vroegrijk, T.: Uniformizable and realcompact bornological universes. Appl. Gen. Topol. 10, 277–287 (2009)
Dedicated to Marco López on the occasion of his 70th birthday.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Beer, G. McShane’s Extension Theorem Revisited. Vietnam J. Math. 48, 237–246 (2020). https://doi.org/10.1007/s10013-019-00366-2
- Uniformly continuous function
- Lipschitz function
- Concave function
- McShane’s Extension Theorem
- Pointwise product of uniformly continuous functions
- Bourbaki bounded set
Mathematics Subject Classification (2010)
- Primary 54C20
- Secondary 26A51