Advertisement

Applied Categorical Structures

, Volume 27, Issue 6, pp 703–721 | Cite as

Neighbourhood Operators: Additivity, Idempotency and Convergence

  • A. RazafindrakotoEmail author
Article
  • 5 Downloads

Abstract

We define and discuss the notions of additivity and idempotency for neighbourhood and interior operators. We then propose an order-theoretic description of the notion of convergence that was introduced by D. Holgate and J. Šlapal with the help of these two properties. This will provide a rather convenient setting in which compactness and completeness can be studied via neighbourhood operators. We prove, among other things, a Frolík-type theorem with the introduction of reflecting morphisms.

Keywords

Neighbourhood operators Interior operators Idempotency Additivity Kleisli composition Kan extension Compactness Convergence Filters 

Mathematics Subject Classification

18B30 54B30 54C10 18B35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Bentley, H.L., Herrlich, H., Lowen, R.: Improving constructions in topology. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, pp. 3–20. Heldermann Verlag, Berlin (1991)Google Scholar
  2. 2.
    Bentley, H.L., Herrlich, H.: Doitchinov’s construct of supertopological spaces is topological. Serdica Math. J. 24(1), 21–24 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bourbaki, N.: General Topology Part 1 (A translation of Éléments de Mathématique, Topologie Générale, originally published in French by Hermann, Paris), vol. 437. Addison-Wesley Pub., Boston (1966)Google Scholar
  4. 4.
    Castellini, G.: Categorical Closure Operators, vol. 300. Birkhäuser, Basel (2003)CrossRefGoogle Scholar
  5. 5.
    Castellini, G.: Interior operators in a category: idempotency and heredity. Topol. Appl. 158, 2332–2339 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Castellini, G.: Interior operators, open morphisms and the preservation property. Appl. Categ. Struct. 23(3), 311–322 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Castellini, G., Murcia, E.: Interior operators and topological separation. Topol. Appl. 160, 1476–1485 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Clementino, M.M., Giuli, E., Tholen, W.: Topology in a category: compactness. Portugaliae Math. 53(4), 1–37 (1996)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Clementino, M.M., Giuli, E., Tholen, W.: A functional approach to general topology. In: Pedicchio, M.C., Tholen, W. (eds.) Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge Univ. Press, Cambridge (2004)Google Scholar
  10. 10.
    Császár, \(\dot{\text{A}}\).: Foundations of General Topology (A translation by Mrs. K. Császár of Fondements de la topologie générale, originally published in French by Akadémiai Kiadó (1960)), 380. Pergamon Press, Oxford (1963)Google Scholar
  11. 11.
    Dikranjan, D.: Semiregular closure operators and epimorphisms in topological categories. In: V International Meeting on Topology (Lecce, 1990/Otranto, 1990). Rend. Circ. Mat. Palermo (2) Suppl. no. 29, pp. 105–160 (1992)Google Scholar
  12. 12.
    Dikranjan, D., Giuli, E.: Closure operators. I. Topol. Appl. 27(2), 129–143 (1987)CrossRefGoogle Scholar
  13. 13.
    Dikranjan, D., Künzi, H.-P.: Separation and epimorphisms in quasi-uniform spaces. Papers in honour of B. Banaschewski (Cape Town, 1996). Appl. Categ. Struct. 8(1–2), 175–207 (2000)CrossRefGoogle Scholar
  14. 14.
    Dikranjan, D., Künzi, H.-P.: Cowellpowerdness of some categories of quasi-uniform spaces. Appl. Categ. Struct. 26(6), 1159–1184 (2018)CrossRefGoogle Scholar
  15. 15.
    Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators, vol. 356. Kluwer Academic Publishers, Dordrecht (1996)zbMATHGoogle Scholar
  16. 16.
    Doitchinov, D.: A unified theory of topological spaces, proximity spaces and uniform spaces. Dokl. Acad. Nauk SSSR 156, 21–24 (1964). (Russian)MathSciNetGoogle Scholar
  17. 17.
    Doitchinov, D.: Supertopological spaces and extensions of topological spaces. Pliska Stud. Math. Bulg. 6, 105–120 (1983). (Russian)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Dydak, J., Weighill, T.: Extension theorems for large scale spaces via coarse neighbourhoods. Mediterr. J. Math. 15(59), 1–28 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Giuli, E., Šlapal, J.: Neighbourhoods with respect to a categorical closure operators. Acta Math. Hung. 124(1–2), 1–14 (2009)CrossRefGoogle Scholar
  20. 20.
    Harris, D.: Extension closed and cluster closed subspaces. Can. J. Math. 24, 1132–1136 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Holgate, D., Iragi, M., Razafindrakoto, A.: Topogenous and nearness structures on categories. Appl. Categ. Struct. 24, 447–455 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Holgate, D., Razafindrakoto, A.: Interior and neighbourhood. Topol. Appl. 168, 144–152 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Holgate, D., Razafindrakoto, A.: A lax approach to neighbourhood operators. Appl. Categ. Struct. 25(3), 431–445 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Holgate, D., Šlapal, J.: Categorical neighbourhood operators. Topol. Appl. 158, 2356–2365 (2011)CrossRefGoogle Scholar
  25. 25.
    Kent, D.C., Min, W.K.: Neighbourhood spaces. IJMMS 32(7), 387–399 (2002)zbMATHGoogle Scholar
  26. 26.
    Leseberg, D.: Superconvergence spaces and related topics. In: Recent Developments of General Topology and Its Applications (Berlin, 1992). Mathematical Research, vol. 67, pp. 187–196. Akademie-Verlag, Berlin (1992)Google Scholar
  27. 27.
    Leseberg, D.: On topologically induced \(b\)-convergences. Topol. Proc. 37, 293–313 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Lowen, R., Tholen, W., Seal, G.J., Hofmann, D., Lucyshyn-Wright, R., Clementino, M.M., Colebunders, E. Lax Algebras (in: Monoidal Topology). In: Hofmann, D., Seal, G.J., Tholen, W. (eds.) Monoidal Topology, 503. Encyclopedia of Mathematics and its Applications, vol. 153. Cambridge Univ. Press, Cambridge (2014) Google Scholar
  29. 29.
    Lunna-Torres, J., Ochoa, C.: Interior operators and topological categories. Adv. Appl. Math. Sci. 10, 189–206 (2011)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 314, 2nd edn. Springer, Berlin (1998)zbMATHGoogle Scholar
  31. 31.
    Reiterman, J., Tholen, W.: Effective descent maps of topological spaces. Topol. Appl. 57, 53–69 (1994)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Richmond, T., Šlapal, J.: Neighbourhood spaces and convergence. Topol. Proc. 35, 165–175 (2010)zbMATHGoogle Scholar
  33. 33.
    Šlapal, J.: Convergence on categories. Appl. Categ. Struct. 16, 503–517 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Šlapal, J.: Compactness and convergence with respect to a neighbourhood operator. Collect. Math. 63, 123–137 (2012)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tholen, W.: A categorical guide to separation, compactness and perfectness. Homol. Homotopy Appl. 1, 147–161 (1999)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tozzi, A., Wyler, O.: On categories of supertopological spaces, 15th winter school in abstract analysis (Srní, 1987). Acta Univ. Carolin. Math. Phys. 28(2), 137–149 (1987)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Vorster, S.J.R.: Interior operators in general categories. Quaest. Math. 23, 405–416 (2000)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wyler, O.: On convergence of filters and ultrafilters to subsets. In: Categorical Methods in Computer Science (Berlin, 1988). Lecture Notes in Computer Science, vol. 393, pp. 340–350. Springer, Berlin (1989) CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of the Western CapeBellvilleSouth Africa

Personalised recommendations