Skip to main content
SpringerLink
Log in

Convergence on Categories

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We introduce and study a concept of a convergence structure on a concrete category. The concept is based on using certain generalized filters for expressing the convergence. Some basic properties of the convergence structures are discussed. In particular, we study convergence separation and convergence compactness and investigate relationships between the convergence structures and the usual closure operators on categories.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990)

    MATH  Google Scholar 

  2. Castellini, G.: Connectedness with respect to a closure operator. Appl. Categ. Structures 9, 285–302 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Castellini, G.: Categorical Closure Operators. Birkhauser, Boston (2003)

    MATH  Google Scholar 

  4. Castellini, G., Giuli, E.: Closure operator with respect to a functor. Appl. Categ. Structures 9, 525–537 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Castellini, G., Hájek, D.: Closure operators and connectedness. Topology Appl. 55, 29–45 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Čech, E.: Topological Spaces, Revised by Z.Frolík and M.Katětov. Academia, Prague, 1966

  7. Clementino, M.M.: On connectedness via closure operator. Appl. Categ. Structures 9, 539–556 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clementino, M.M., Giuli, E., Tholen, W.: Topology in a category: compactness. Portugal. Math. 53, 397–433 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Clementino, M.M, Giuli, E., Tholen, W.: What is a quotient map with respect to a closure operator? Appl. Categ. Structures 9, 139–151 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clementino, M.M., Tholen, W.: Tychonoff’s Theorem in a category. Proc. Amer. Math. Soc. 124, 3311–3314 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Clementino, M.M., Tholen, W.: Separation versus connectedness. Topology Appl. 75, 143–181 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dikranjan, D., Giuli, E.: Closure operators I. Topology Appl. 27, 129–143 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dikranjan, D., Giuli, E.: Compactness, minimality and closedness with respect to a closure operator. In: Proceedings of the International Conference on Categorical Topology, pp. 279–335. Prague 1988, World Scientific, Singapore (1989)

  14. Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Giuli, E., Tholen, W.: Openness with respect to a closure operator. Appl. Categ. Structures 8, 487–502 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Giuli, E., Šlapal, J.: Raster convergence with respect to a closure operator. Cahiers Topologie Géom. Différentielle Catég. 46, 275–300 (2005)

    MATH  Google Scholar 

  18. Grätzer, G.: General Lattice Theory. Birkhäuser Verlag, Basel (1978)

    Google Scholar 

  19. Preuss, G.: Semiuniform convergence spaces. Math. Japonica 41, 465–491 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Šlapal, J.: Net spaces in categorical topology. Ann. N.Y. Acad. Sci. 806, 393–412 (1996)

    Article  Google Scholar 

  21. Šlapal, J.: Compactness in categories of S-net spaces. Appl. Categ. Structures 6, 515–525 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Šlapal, J.: Compactification of limit \(\mathcal S\)-net spaces. Monatsh. Math. 131, 335–341 (2000)

    Article  MathSciNet  Google Scholar 

  23. Šlapal, J.: Convergence structures for categories. Appl. Categ. Structures 9, 557–570 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Šlapal, J.: Compactness with respect to a convergence structure. Quaestiones Math. 25, 19–27 (2002)

    MATH  Google Scholar 

  25. Šlapal, J.: F-net convergence with respect to a closure operator. Appl. Categ. Structures 13, 49–64 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brno University of Technology, 616 69, Brno, Czech Republic

    Josef Šlapal

Authors
  1. Josef Šlapal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Josef Šlapal.

Additional information

Dedicated to Professor E. Giuli and Professor G. Preuss on the occasion of their 65th birthdays.

Financial support from the Ministry of Education of the Czech Republic (research plan MSM0021630518) is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Šlapal, J. Convergence on Categories. Appl Categor Struct 16, 503–519 (2008). https://doi.org/10.1007/s10485-007-9109-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-007-9109-0

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation