Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The category of $ D$-completely regular spaces is simple

Author: N. C. Heldermann
Journal: Trans. Amer. Math. Soc. 262 (1980), 437-446
MSC: Primary 54D30; Secondary 18B30
MathSciNet review: 586727
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent paper H. Brandenburg characterized the objects of the epireflective hull of all developable spaces-that are those spaces which are homeomorphic to a subspace of a product of developable spaces-by intrinsic properties. It is shown here that these spaces, called D-completely regular, can be generated from a single second countable developable space D which has the same cardinality as the reals. As an application of this result we obtain a new characterization of D-normal spaces analogous to Urysohn's lemma and a new (external) characterization of perfect spaces (meaning every closed set is a $ {G_\delta }$).

References [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Metrization of developable spaces, Canad. J. Math 3 (1951), 175-186. MR 0043449 (13:264f)
  • [2] H. Brandenburg, Hüllenbildungen für die Klasse der entwickelbaren topologischen Räume, Dissertation, Freie Universität, Berlin, 1979.
  • [3] -, On a class of nearness spaces and the epireflective hull of developable topological spaces, Proc. Topology Conf., Belgrad, 1977 (to appear).
  • [4] -, Die epireflektive Hülle der entwickelbaren topologischen Räume, Nordwestdeutsches Kategorienseminar (Bericht der Tagung in Bielefeld, 1977), G. Richter, Ed., Bielefeld, 1978, pp. 33-49.
  • [5] -, On spaces with a $ {G_\delta }$-basis, Arch. Math. (Basel) (to appear).
  • [6] -, Some characterizations of developable spaces, Proc. Amer. Math. Soc. 80 (1980), 157-161. MR 574527 (81g:54027)
  • [7] -, On E-normal spaces (Proc. Conf. Categorical Topology, Berlin, 1978), H. Herrlich and G. Preuss, Eds., Lecture Notes in Math., vol. 719, Springer-Verlag, Berlin and New York, 1979, pp. 24-34. MR 544628 (81d:54013)
  • [8] -, Separating closed sets by continuous mappings into developable spaces, in preparation.
  • [9] R. Engelking and S. Mrowka, On E-compact spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 6 (1958), 429-436. MR 0097042 (20:3522)
  • [10] N. C. Heldermann, Developability and some new regularity axioms. Canad. J. Math. (to appear). MR 627648 (82m:54013)
  • [11] H. Herrlich and G. E. Strecker, Category theory: an introduction, 2nd ed., Sigma Series in Pure Math., Vol. 1, Heldermann Verlag, Berlin, 1979. MR 571016 (81e:18001)
  • [12] T. Marny, Rechts-Bikategoriestrukturen in topolologischen Kategorien, Dissertation, Freie Universität, Berlin, 1973.
  • [13] -, On epireflective subcategories of topologiccal categories, General Topology Appl. 10 (1979), 175-181. MR 527843 (80g:18004)
  • [14] S. Mrowka, Further results on E-compact spaces. I, Acta Math. 120 (1968), 161-185. MR 0226576 (37:2165)
  • [15] A. Mysior, On generalized classes of complete regularity, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 341-342. MR 0405349 (53:9143)
  • [16] J. M. Worrell, Jr., Upper semicontinuous decompositions of developable spaces, Proc. Amer. Math. Soc. 16 (1965), 485-490. MR 0181982 (31:6207)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54D30, 18B30

Retrieve articles in all journals with MSC: 54D30, 18B30

Additional Information

Keywords: Developable spaces, D-completely regular spaces, D-normal spaces, epireflective hull, simple category, perfect spaces
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society