Log in

Uniform density topology

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Main results of the paper are: the Lebesgue Density Theorem does not hold for the uniform density points and the uniform density topology is completely regular.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bruckner, Andrew M.: Differentiation of Real Functions, Lecture Notes in Mathematics 659. Springer Verlag (1978)

    Book  Google Scholar 

  2. Brown, Tom C., Freedman, Allen R.: The uniform density of sets of integers and Fermat’s Last Theorem. C. R. Math. Rep. Acad. Sci. Can. 12, 1–6 (1990)

    ADS  MathSciNet  Google Scholar 

  3. Denjoy, Arnaud: Mémoire sur les dérivés des founction continues. J. Math. Pures et Appl. 1, 105–240 (1915)

    Google Scholar 

  4. Goffman, Casper, Neugebauer, C.J., Nishiura, Togo: The density topology and approximate continuity. Duke Math. J. 28, 497–506 (1961)

    Article  MathSciNet  Google Scholar 

  5. Goffman, Casper, Waterman, Daniel: Approximately continuous transformations. Proc. Am. Math. Soc. 12, 116–121 (1961)

    Article  MathSciNet  Google Scholar 

  6. Haupt, O.: Pauc, Christian; La topologie de Denjoy envisagée comme vraie topologie. C. R. Acad. Sci. Paris 234, 390–392 (1952)

    MathSciNet  Google Scholar 

  7. Hejduk, Jacek, Wiertelak, Renata: On the abstract density topologies generated by lower and almost lower density operators, Traditional and present-day topics in real analysis, Dedicated to Professor Jan Stanisław Lipiński, Łódź University Press, 431-447 (2013)

  8. Oxtoby, J.C.: Measure and Category, Springer Verlag (1971)

  9. Taylor, S.J.: On strengthening the Lebesgue density theorem. Fund. Math. 46, 305–315 (1959)

    Article  MathSciNet  Google Scholar 

  10. Tall, Franklin D.: The density topology. Pacific Math. J. 62(1), 275–284 (1976)

    Article  MathSciNet  Google Scholar 

  11. Wilczyński, Władysław: Density topologies. Chapter 15 in Handbook of Measure Theory. Edited by E. Pap. Elsevier, 675-702 (2002)

  12. Wilczyński, W.ładysław: On the Lebesgue density theorem. J. Appl. Anal. 18, 275–281 (2012)

    Article  MathSciNet  Google Scholar 

  13. Wilczyński, W.ładysław: A category analogue of the density topology non-homeomorphic with the \(I\)-density topology. Positivity 23, 469–484 (2019)

    Article  MathSciNet  Google Scholar 

  14. Wilczyński, W.ładysław, Wojdowski, Łukasz, Wojdowski, Wojciech: Points of density ideals of subsets of \(\textbf{N} \). Georgian Math. J. 26(4), 529–535 (2019)

    Article  MathSciNet  Google Scholar 

  15. Zahorski, Z.: Sur le première dérivée. Trans. Am. Math. Soc. 69(1), 1–54 (1950)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Władysław Wilczyński.

Additional information

Communicated by Michael Kunzinger.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

All data is provided in full in the results section of this paper

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wilczyński, W. Uniform density topology. Monatsh Math 203, 717–732 (2024). https://doi.org/10.1007/s00605-023-01929-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-023-01929-9

Keywords

Keywords

Mathematics Subject Classification

Navigation