[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Covering Dimension and Universality Property on Frames

  • Published:
Order 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

Studies such as Dube et al. (Topology Appl. 160(18):2454–2464 2013), Georgiou et al. (Algebra Universalis 80(2):16, 2019), Iliadis (2005; Topology Appl. 179, 99–110, 2015; Topology Appl. 201, 92–109, 2016) focus on the notion of saturated class of spaces and frames and the existence of universal elements in such classes. Recently, the notion of dimension for frames has been combined with this universality property (Georgiou et al., Algebra Universalis 80(2):16, 2019; Iliadis, Topology Appl. 201:92–109, 2016). In this paper we study such a property, combining it with the notion of the covering dimension dim for frames (Charalambous, J. London Math. Soc. 8(2):149–160, 1974). We define the base dimension like-function of the type dim for frames and, based on the notion of the saturated class of bases, which is inserted and studied in Georgiou et al. (Order 37:427–444, 2020), we prove that in a class of bases which is characterized by this dimension there exist universal elements. Also, we study the notion of saturated class of frames, which is inserted in Georgiou et al. (Algebra Universalis 80(2):16, 2019), proving that in classes of frames which are determined by the covering dimension dim there exist universal elements.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /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 (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Banaschewski, B., Gilmour, G.: Stone-Čech compactification and dimension theory for regular σ-frames. J. London Math. Soc. 39(2), 1–8 (1989)

    Article  MathSciNet  Google Scholar 

  2. Banaschewski, B.: Universal Zero-Dimensional Compactifications. Categorical Topology and its Relation to Analysis, Algebra and Combinatorics (Prague, 1988), pp 257–269. World Sci. Publ., Teaneck (1989)

    Google Scholar 

  3. Brijlall, D., Baboolal, D.: Some aspects of dimension theory of frames. Indian J. Pure Appl. Math. 39(5), 375–402 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Brijlall, D., Baboolal, D.: The Katětov-Morita theorem for the dimension of metric frames. Indian J. Pure Appl. Math. 41(3), 535–553 (2010)

    Article  MathSciNet  Google Scholar 

  5. Charalambous, M. G.: Dimension theory of σ-frames. J. London Math. Soc. 8(2), 149–160 (1974)

    Article  MathSciNet  Google Scholar 

  6. Charalambous, M. G.: The dimension of paracompact normal κ-frames. Topology Proc. 20, 49–66 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Charalambous, M. G.: Dimension Theory, a Selection of Theorems and Counterexamples Atlantis Studies in Mathematics, vol. 7. Springer International Publishing, Switzerland (2019)

    Book  Google Scholar 

  8. Dube, T., Iliadis, S., van Mill, J., Naidoo, I.: Universal frames. Topology Appl. 160(18), 2454–2464 (2013)

    Article  MathSciNet  Google Scholar 

  9. Engelking, R.: Theory of dimensions, finite and infinite. Sigma Series in Pure Mathematics 10 Lemgo (1995)

  10. Español, L., Gutiérrez García, J., Kubiak, T.: Separating families of locale maps and localic embeddings. Algebra Universalis 67(2), 105–112 (2012)

    Article  MathSciNet  Google Scholar 

  11. Georgiou, D., Iliadis, S., Megaritis, A., Sereti, F.: Small inductive dimension and universality on frames. Algebra Universalis 80(2), 16 (2019). Art. 21

    Article  MathSciNet  Google Scholar 

  12. Georgiou, D., Iliadis, S., Megaritis, A., Sereti, F.: Universality property and dimension for frames. Order 37, 427–444 (2020)

    Article  MathSciNet  Google Scholar 

  13. Georgiou, D., Kougias, I., Megaritis, A., Prinos, G., Sereti, F.: A study of a new dimension for frames. Topology Appl. 275, 106995 (2020)

    Article  MathSciNet  Google Scholar 

  14. Gevorgyan, P. S., Iliadis, S. D., Sadovnichy, Y.V.: Universality on frames. Topology Appl. 220, 173–188 (2017)

    Article  MathSciNet  Google Scholar 

  15. Iliadis, S. D.: Universal spaces and mappings North-Holland mathematics studies, vol. 198. Elsevier Science B.V., Amsterdam (2005)

    Google Scholar 

  16. Iliadis, S. D.: Universal regular and completely regular frames. Topology Appl. 179, 99–110 (2015)

    Article  MathSciNet  Google Scholar 

  17. Iliadis, S. D.: Dimension and universality on frames. Topology Appl. 201, 92–109 (2016)

    Article  MathSciNet  Google Scholar 

  18. Isbell, J. R.: Graduation and dimension in locales, Aspects of topology, 195–210. London Math. Soc. Lecture Note Ser. 93. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  19. Liu, Y.-M., Luo, M-K.: Fuzzy Topology Advances in Fuzzy Systems – Applications and Theory, vol. 9. World Scientific Publishing Co., Inc., River Edge (1997)

    Google Scholar 

  20. Pears, A. R.: Dimension Theory of General Spaces. Cambridge University Press, Cambridge (1975)

    MATH  Google Scholar 

  21. Picado, J., Pultr, A.: Frames and Locales, Topology Without Points. Frontiers in Mathematics. Birkhäuser/Springer, Basel (2012)

    MATH  Google Scholar 

  22. Pultr, A., Tozzi, A.: Equotionally closed subframes and represetations of quotient spaces. Cahiers Topologie géom Différentielle Catég. 34(3), 167–183 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Sancho de Salas, J. B., Sancho de Salas, M. T.: Dimension of distributive lattices and universal spaces. Topology Appl. 42(1), 25–36 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referee for the careful reading of the paper.

Author information

Author notes

  1. T. Dube

    Present address: Department of Mathematical Sciences, University of South Africa, Pretoria, 0003, South Africa

Authors and Affiliations

  1. Department of Mathematical Sciences, University of South Africa, Pretoria, 0003, South Africa

    D. Georgiou & I. Naidoo

  2. Department of Mathematics, University of Patras, Patras, 26500, Greece

    D. Georgiou

  3. Department of Physics, University of Thessaly, Lamia, 35100, Greece

    A. Megaritis

  4. Department of Museology, University of Patras, Pyrgos, 27131, Greece

    F. Sereti

Authors
  1. T. Dube
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Georgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Megaritis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. I. Naidoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. F. Sereti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Georgiou.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dube, T., Georgiou, D., Megaritis, A. et al. Covering Dimension and Universality Property on Frames. Order 39, 187–208 (2022). https://doi.org/10.1007/s11083-021-09561-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-021-09561-8

Keywords

Search

Navigation