Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T01:55:47.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Amsterdam Properties of Cp(X) Imply Discreteness of X

Published online by Cambridge University Press:  20 November 2018

D. J. Lutzer ,
J. van Mill  and
V. V. Tkachuk
D. J. Lutzer
Affiliation:
Mathematics Department, College of William and Mary, Williamsburg, VA 23187-8795, U.S.A.. e-mail: lutzer@math.wm.edu
J. van Mill
Affiliation:
Department of Mathematics, Vrije Universiteit, HV Amsterdam, The Netherlands. e-mail: vanmill@cs.vu.nl
V. V. Tkachuk
Affiliation:
Departamento de Matematicas, Universidad Autónoma Metropolitana, Mexico, D.F.. e-mail: vova@xanum.uam.mx
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 prove, among other things, that if ${{C}_{p}}\left( X \right)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

Keywords

54B1054C0554D30regular filterbasesubcompact spacefunction spacediscrete space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 4 , 01 December 2008 , pp. 570 - 578
Copyright
Copyright © Canadian Mathematical Society 2008

References

[AL] Aarts, J. M. and Lutzer, D. J., Completeness properties designed for recognizing Baire spaces. Dissertationes Math. 116(1974), 1–48.Google Scholar
[Ar] Arhangel’skii, A. V., Topological Function Spaces. Mathematics and its Applications 78, Kluwer, Dordrecht, 1992.Google Scholar
[Ch] Choquet, G., Lectures on Analysis. Benjamin, New York, 1969.Google Scholar
[DGLM] Dijkstra, J., Grillot, T., Lutzer, D. J., and van Mill, J., Function spaces of the low Borel complexity. Proc Amer. Math. Soc. 94(1985), no. 4, 703–710.Google Scholar
[En] Engelking, R., General Topology. PWN–Polish Scientific Publishers, Warszawa 1977.Google Scholar
[Ju] Juhász, I., Cardinal Functions in Topology—Ten Years Later. Second edition. Mathematical Centre Tracts 123, Mathematisch Centrum, Amsterdam, 1980.Google Scholar
[LM] Lutzer, D. J. and McCoy, R. A., Category in function spaces. I. Pacific J. Math., 90(1980), no. 1, 145–168.Google Scholar
[Ox] Oxtoby, J., Cartesian products of Baire spaces. Fund. Math. 49(1960/1961), 157–166.Google Scholar
[Tk1] Tkachuk, V. V., Duality with respect to the functor Cp and cardinal invariants of the type of the Suslin number. Math. Notes 37(1985), no. 3, 247–252.Google Scholar
[Tk2] Tkachuk, V. V., The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties. Topology Appl. 22(1986), no. 3, 241–253.Google Scholar
[Tk3] Tkachuk, V. V., Mapping metric spaces and their products onto Cp(X). New Zealand J. Math. 27(1998), no. 1, 113–122.Google Scholar
[vM] van Mill, J., Cp(X) is not Gδσ: a simple proof. Bull. Polon. Acad. Sci. Math. 47(1999), no. 4, 319–323.Google Scholar