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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Bentley, H.L., Herrlich, H.: Countable choice and pseudometric spaces. In: 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). Topology and its Applications, vol. 85, no. (1–3), pp. 153–164 (1998)
Herrlich, H.: Axiom of Choice. Lecture Notes in Mathematics, vol. 1876. Springer, Berlin (2006)
Herrlich, H., Keremedis, K.: On the metric reflection of a pseudometric space in $\mathbf{ZF}$. Comment. Math. Univ. Carolinae 56(1), 77–88 (2015)
Howard, P., Rubin, J.E.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs, vol. 59. American Mathematical Society, Providence (1998)
Keremedis, K.: On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice. Topol. Appl. 108(1), 75–78 (2000)
Keremedis, K.: On metric spaces where continuous real valued functions are uniformly continuous in $\mathbf{ZF}$. Topol. Appl. 210, 366–375 (2016)
Keremedis, K.: Uniform continuity and normality of metric spaces in ZF. Bull. Pol. Acad. Sci. Math. 65, 113–124 (2017)
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.
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Funding was provided by FAPESB (Grant No. APP0072/2016).
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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