McShane’s Extension Theorem Revisited

Abstract

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.

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Correspondence to Gerald Beer.

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Dedicated to Marco López on the occasion of his 70th birthday.

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Beer, G. McShane’s Extension Theorem Revisited. Vietnam J. Math. 48, 237–246 (2020). https://doi.org/10.1007/s10013-019-00366-2

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Keywords

  • 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
  • 26A16
  • 54C30
  • Secondary 26A51
  • 54E40