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On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice

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Abstract

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between (pseudo)metric spaces, and the second declares that sequentially compact pseudometric spaces are \(\mathbf {UC}\)—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

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Acknowledgements

The author is grateful to his colleague Marcelo D. Passos for calling his attention to characterizations of uniform continuity of real valued functions which do not rely on the Lebesgue covering number. The author also acknowledges the anonymous referee for his/her careful reading of the manuscript and for a number of helping comments and corrections which improved the presentation of the paper.

Funding

Funding was provided by FAPESB (Grant No. APP0072/2016).

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia Campus Ondina, Av. Adhemar de Barros, S/N, Ondina, Salvador, BA, 40170-110, Brazil

    Samuel G. da Silva

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  1. Samuel G. da Silva
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Correspondence to Samuel G. da Silva.

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This paper is dedicated to the memory of Prof. Horst Herrlich (1937–2015).

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da Silva, S.G. On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice. Arch. Math. Logic 58, 353–358 (2019). https://doi.org/10.1007/s00153-018-0643-2

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  • DOI: https://doi.org/10.1007/s00153-018-0643-2

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