Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-04T16:36:34.752Z Has data issue: false hasContentIssue false
  • English
  • Français

Simplicity of Categories Defined by Symmetry Axioms

Published online by Cambridge University Press:  20 November 2018

E. Lowen-Colebunders  and
Z. G. Szabo
E. Lowen-Colebunders
Affiliation:
Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel
Z. G. Szabo
Affiliation:
Department for Analysis, L. Eötvös University Budapest, Muzeum krt 6-8, H-1088 Budapest
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.

Keywords

AMS (1980) Subject classification: 54A0554B30.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 2 , 01 June 1991 , pp. 240 - 248
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bentley, H. L., Herrlich, H. and Robertson, W. A., Convenient Categories for topologists, Comment. Math. Univ. Carol. 17(1976), 207–227.Google Scholar
2. Bentley, H. L., Herrlich, H. and Lowen-Colebunders, E., Convergence, J. Pure and App. Alg. 65(1990), 27–45.Google Scholar
3. Binz, E., Continuous Convergence on C(X), Lect. Notes in Math. 469, Springer-Verlag, Berlin, 1975.Google Scholar
4. Fisher, H. R., Limesraume, Math. Ann. 137(1959), 269–303.Google Scholar
5. Gazik, R. and Kent, D. C., Regular completions ofCauchy spaces via function algebras, Bull. Austral. Math. Soc. 11(1974), 77–88.Google Scholar
6. Hajek, D. W. and Mysior, A., On non-simplicity of topological categories, Lect. Notes in Math. 719(1979), 84–93.Google Scholar
7. Herrlich, H., Wann sind aile stetigen Abbildungen in Y konstant? Math. Zeitschr. 90(1965), 152–154.Google Scholar
8. Herrlich, H., Topological structures, in: Topological structures, Math. Centre Tracts 52, Amsterdam 1974, 59–122.Google Scholar
9. Herrlich, H., Categorical topology 1971-1981, in: Proc. of the Fifth Prague Topological Symposium 1981, Heldermann Verlag, Berlin, 1983, 279–383.Google Scholar
10. Katetov, M., On continuity structures and spaces of mappings, Comment. Math. Univ. Carol. 6(1965), 257–278.Google Scholar
11. Kent, D. C., McKennon, K., G. Richardson and Schroder, M., Continuous convergence in C﹛X), Pacific J. Math. 52(1974), 271–279.Google Scholar
12. Lowen-Colebunders, E., Function classes of Cauchy Continuous Maps, Pure and Applied Mathematics, Marcel Dekker Inc., New York, 1989.Google Scholar
13. Lowen-Colebunders, E. and Szabo, Z. G., On the simplicity of some categories of closure spaces, comment. Math. Univ. Carol. 31(1990), 95–98.Google Scholar
14. Lowen, E. and Lowen, R., On the non-simplicity of some convergence categories, Proc. Amer. Math. Soc, 105(1989), 305–308.Google Scholar
15. Marny, T., Rechts-Bikategoriestructuren in topologischen Kategorien, Thesis, F.U. Berlin, 1973.Google Scholar
16. Robertson, W. A., Convergence as a Nearness Concept, Thesis, Carleton University, 1975.Google Scholar