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On metric spaces with non-equivalent Hewitt and Samuel realcompactifications

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Let D be a uniformly discrete space, let \(\pi \) be the product uniformity on the countable power \(D^{\mathbb {N}}\) of D and let \(e\pi \) be the uniformity on \(D^{\mathbb {N}}\) induced by all the countable covers in \(\pi \). Assume that the cardinality of D is Ulam measurable. Then \((D^{\mathbb {N}}, e\pi )\) has a Cauchy filterbase, consisting of closed sets, which is not countably centered. As a consequence, the Hewitt and Samuel realcompactifications of \((D^{\mathbb {N}},\pi )\) are not equivalent.

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References

  1. Bourbaki, N.: Elements of Mathematics, General Topology, Part 1. Hermann, Paris (1966)

    MATH  Google Scholar 

  2. Chekeev, A.A.: Uniformities for Wallman compactifications and realcompactifications. Topol. Appl. 201, 145–156 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  4. Comfort, W.W., Negrepontis, S.: Continuous Pseudometrics. Marcel Dekker, New York (1975)

    MATH  Google Scholar 

  5. Curzer, H., Hager, A.W.: Zero-set ultrafilters and \(G_{\delta }\)-closures in uniform spaces. J. Aust. Math. Soc. 28, 219–228 (1979)

    Article  MATH  Google Scholar 

  6. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  7. Garrido, M.I., Meroño, A.S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39, 733–758 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garrido, M.I., Meroño, A.S.: The Samuel realcompactification of a metric space. J. Math. Anal. Appl. 456, 1013–1039 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garrido, M.I., Meroño, A.S.: The Samuel realcompactification. Topol. Appl. 241, 150–161 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gillman, L., Jerison, M.: Rings of Continuous Functions. D. Van Nostrand Co, New Jersey (1960)

    Book  MATH  Google Scholar 

  11. Hager, A.W.: On inverse-closed subalgebras of \(C(X)\). Proc. Lond. Math. Soc. 19, 233–257 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Howes, N.R.: Modern Analysis and Topology. Spinger, New York (1995)

    Book  MATH  Google Scholar 

  13. Hušek, M.: Variations of Uniform completeness related to realcompactness. Comment. Math. Univ. Carol. 58, 501–518 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Isbell, J.R.: Euclidean and weak uniformities. Pac. J. Math. 8, 67–86 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jech, T.: Set Theory. The Third Millennium Edition Revised and Expanded. Springer, Heidelberg (2003)

    Google Scholar 

  16. Meroño, A.S.: Bourbaki-completeness and Samuel Realcompactification, Ph.D. Thesis (2019)

  17. Reynolds, G.D., Rice, M.D.: Completeness and covering properties of uniform spaces. Q. J. Math. Oxf. 29, 367–374 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Steiner, A.K., Steiner, E.F.: Nest generated intersection rings in Tychonoff spaces. Trans. Am. Math. Soc. 148, 589–601 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  19. Willard, S.: General Topology. Dover Publications, Minneola (2004)

    MATH  Google Scholar 

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Acknowledgements

We would like to thank Miroslav Hušek and Aarno Hohti for their valuable comments on the problem that has been solved in this paper.

Funding

This study was partially supported by Universidad Complutense de Madrid, Ministerio de Economía y Competitividad—PGC2018-097286-B-I00.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, 00140, Helsinki, Finland

    Heikki Junnila

  2. Departamento de Matemática Aplicada a la Ingeniería Industrial, E.T.S. de Ingenieros Industriales, Universidad Politécnica de Madrid, 28006, Madrid, Spain

    Ana S. Meroño

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  1. Heikki Junnila
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  2. Ana S. Meroño
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Correspondence to Ana S. Meroño.

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Junnila, H., Meroño, A.S. On metric spaces with non-equivalent Hewitt and Samuel realcompactifications. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 54 (2023). https://doi.org/10.1007/s13398-022-01384-5

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  • DOI: https://doi.org/10.1007/s13398-022-01384-5

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