Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An internal and an external characterization of convergence spaces in which adherences of filters are closed

Author: Eva Lowen-Colebunders
Journal: Proc. Amer. Math. Soc. 72 (1978), 205-210
MSC: Primary 54A05; Secondary 54A20
MathSciNet review: 0500785
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this note is to give necessary and sufficient conditions for a convergence space (X, q) such that for every filter on X its adherence is a closed subset of (X, q). An internal characterization of this property is given by weakening the diagonal condition of Kowalsky. An external characterization is given using a hyperspace structure on the collection of all closed subsets of the given space. It will be shown that a convergence space has closed adherences if and only if the hyperspace is compact.

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

  • [1] N. Bourbaki, Topologie générale, 4th ed., Éléments de mathématique, Fasc. II, Chap. 1, Hermann, Paris, 1965. MR 0244924 (39:6237)
  • [2] G. Choquet, Convergences, Ann. Univ. Grenoble Sect. Sci. Math. Phys. 23 (1948), 57-112. MR 0025716 (10:53d)
  • [3] H. Fisher, Limesraüme, Math. Ann. 137 (1959), 269-303.
  • [4] Z. Frolík, Concerning topological convergence of sets, Czechoslovak Math. J. 10 (1960), 168-180. MR 0116303 (22:7098)
  • [5] D. Kent, Convergence functions and their related topologies, Fund. Math. 54 (1964), 125-133. MR 0161301 (28:4509)
  • [6] D. Kent and G. Richardson, Regular completions of Cauchy spaces, Pacific J. Math. 51 (1974), 483-490. MR 0390989 (52:11811)
  • [7] H. Kowalsky, Limesraüme und Komplettierung, Math. Nachr. 12 (1954), 301-340. MR 0073147 (17:390b)
  • [8] E. Lowen-Colebunders, The Choquet hyperspace structure for a convergence space, (to appear).
  • [9] S. Mrówka, Some comments on the space of subsets, Lecture Notes in Math. 171 (1970), 59-63. MR 0270327 (42:5216)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54A05, 54A20

Retrieve articles in all journals with MSC: 54A05, 54A20

Additional Information

Keywords: Convergence spaces, closed adherences, almost topological spaces, diagonal spaces, hyperspace
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society