Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-08T20:47:36.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Developability and Some New Regularity Axioms

Published online by Cambridge University Press:  20 November 2018

N. C. Heldermann
N. C. Heldermann*
Affiliation:
Zentralblatt für Mathematik, Berlin, Germany
Rights & Permissions [Opens in a new window]

Extract

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.

In a recent publication H. Brandenburg [5] introduced D-completely regular topological spaces as a natural extension of completely regular (not necessarily T1) spaces: Whereas every closed subset A of a completely regular space X and every x ∈ X\A can be separated by a continuous function into a pseudometrizable space (namely into the unit interval), D-completely regular spaces admit such a separation into developable spaces. In analogy to the work of O. Frink [16], J. M. Aarts and J. de Groot [19] and others ([38], [46]), Brandenburg derived a base characterization of D-completely regular spaces, which gives rise in a natural way to two new regularity conditions, D-regularity and weak regularity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 641 - 663
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Armentrout, S., A Moore space on which every real-valued continuous function is constant, Proc. Amer. Math. Soc. 12 (1961), 106–109.Google Scholar
2. Bing, R. H., Metrization of developable spaces, Can. J. Math. 3 (1951), 175–186.Google Scholar
3. Bogin, R. G., Spazi quoziente e proprieta di separazione, Univ. Politec. Torino, Rend. Sem. Mat. 34 (1975-76), 165–173.Google Scholar
4. Brandenburg, H., Hilllenbildungen fur die Klasse der entwickelbaren topologischen Rdume, Dissertation, Freie Universitât Berlin (1979).Google Scholar
5. Brandenburg, H., On a class of nearness spaces and the epirefiective hull of developable topological spaces, Proc. of the Int. Topol. Symp., Belgrade (1977), to appear.Google Scholar
6. Brandenburg, H., Die epireflektive Hiille der entwickelbaren topologischen Rdume, Nordwestdeutsches Kategorienseminar, Bericht der Tagung in Bielefeld (1977), Bielefeld (1978), 33–49.Google Scholar
7. Brandenburg, H., On spaces with a G-basis, to appear.Google Scholar
8. Brandenburg, H., Some characterizations of developable spaces, Proc. Amer. Math. Soc, to appear.Google Scholar
9. Brandenburg, H., Separating closed sets by continuous mappings into developable spaces, submitted.Google Scholar
10. Burke, D. K., On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655–663.Google Scholar
11. Chaber, J., Remarks on open-closed mappings, Fundamenta Math. 74 (1972), 197–208.Google Scholar
12. Chaber, J., On subparacompactness and related properties, General Topol. Appl. 10 (1979), 13–17.Google Scholar
13. Davis, A. S., Indexed systems of neighbourhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893.Google Scholar
14. van Douwen, E., A regular space on which every continuous real-valued function is constant, Nieuw Arch. Wiskunde, III. Ser. 20 (1972), 143–145.Google Scholar
15. Engelking, R. and Mrowka, S., On E-compact spaces, Bull. Acad. Polon. Sci., Ser. Sci. math, astron. phys. 6 (1958), 429–436.Google Scholar
16. Frink, O., Compactifications and semi-normal spaces, Amer. J. Math. 86 (1964), 602–607.Google Scholar
17. Gantner, T. E., A regular space on which every continuous real-valued function is constant, Amer. Math. Monthly 78 (1971), 52–53.Google Scholar
18. Gillman, L. and Jerison, M., Rings of continuous functions, second edition, Graduate Texts in Mathematics 43, New York et al (1976).Google Scholar
19. de Groot, J. and Aarts, J. M., Complete regularity as a separation axiom, Can. J. Math. 21 (1969), 96–105.Google Scholar
20. Heldermann, N. C., The category of D-completely regular spaces is simple, Trans. Amer. Math. Soc, to appear.Google Scholar
21. Henrikson, M. and Isbell, J. R., Some properties of compactifications, Duke Math. J. 25 (1958), 83–106.Google Scholar
22. Herrlich, H., Wann sind allé stetigen Abbildungen in Ykonstant?, Math. Z. 90 (1965), 152–154.Google Scholar
23. Herrlich, H. and Strecker, G. E., Category theory, second edition (Heldermann Verlag, Berlin, 1979).Google Scholar
24. Jones, F. B., Moore spaces and uniform spaces, Proc. Amer. Math. Soc. 9 (1958), 483–486.Google Scholar
25. Kelley, J. L., General topology, Graduate Texts in Mathematics 27, New York et al (1977).Google Scholar
26. Kramer, T. R., A note on countably subparacompact spaces, Pacif.c J. Math. 4® (1973), 209–213.Google Scholar
27. Krenger, P. and Raetz, J., On the sup and inf invariance of some separation axioms, Publ. Inst. Math., Beograd, n. Sér. 22(36) (1977), 145–147.Google Scholar
28. Th., Marny, Rechts-Bikategoriestrukturen in topologischen Kategorien, Dissertation, Freie Universitât Berlin (1973).Google Scholar
29. Misra, A. K., A topological view of P-spaces, General Topol. Appl. 2 (1972), 349–362.Google Scholar
30. Mrowka, S., Further results on E-compact spaces. I, Acta Math. 120 (1968), 161–185.Google Scholar
31. Mysior, A., On generalized classes of complete regularity, Bull. Acad. Polon. Sci., Sér. Sci. math, astron. phys. 24 (1976), 341–342.Google Scholar
32. Mysior, A., Two remarks on D-regular spaces, Glasnik mat., III. Ser., to appear.Google Scholar
33. Mysior, A., A regular space which is not completely regular, preprint.Google Scholar
34. Pol, R. and Puzio-Pol, E., Remarks on cartesian products, Fundamenta math. 93 (1976), 57–69.Google Scholar
35. Roy, P., Dual of a Moore space, Notices Amer. Math. Soc. 9 (1962), 327–328.Google Scholar
36. Shanin, N. A., On separation in topological spaces, Doklady Akad. Nauk SSSR 38 (1943), 110–113.Google Scholar
37. Steen, L. A. and Seebach, J. A., Jr., Counterexamples in topology, second edition, Berlin et al. (1978).Google Scholar
38. Steiner, E. F., Normal families and completely regular spaces, Duke Math. J. 33 (1966), 743–745.Google Scholar
39. Thomas, J., A regular space, not completely regular, Amer. Math. Monthly 76 (1969), 181–182.Google Scholar
40. Thomas, J., Associated regular spaces, Can. J. Math. 20 (1968), 1087–1092.Google Scholar
41. Tsai, J. H., On E-compact spaces and generalizations of perfect mappings, Pacific J. Math. Jfi (1973), 275–282.Google Scholar
42. Tychonoff, A., Tiber die topologische Erweiterung von Rdumen, Math. Ann. 102 (1930), 544–561.Google Scholar
43. Tychonoff, A., Ûber einen Metrisationssatz von P. Urysohn, Math. Ann. 95 (1926), 139–142.Google Scholar
44. Willard, S., General topology (Addison Wesley, Reading, Massachusetts et al., 1970).Google Scholar
45. Younglove, J. N., A locally connected, complete Moore space on which every real-valued continuous function is constant, Proc. Amer. Math. Soc. 20 (1969), 527–530.Google Scholar
46. Zaicev, V. I., On the theory of Tychonoff spaces, Vestnik Moskov. Univ., Ser. I 22 (1967), 48–57 (Russian).Google Scholar