Hostname: page-component-6bf8c574d5-7jkgd Total loading time: 0 Render date: 2025-02-17T14:11:17.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequential Compactness of X Implies a Completeness Property for C(X)

Published online by Cambridge University Press:  20 November 2018

M. Rajagopalan  and
R. F. Wheeler
M. Rajagopalan
Affiliation:
Memphis State University, Memphis, Tennessee
R. F. Wheeler
Affiliation:
Memphis State University, Memphis, Tennessee
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.

A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 1 , 01 February 1976 , pp. 207 - 210
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Brauner, K., A class of topologically free locally convex spaces and related group Hopf algebras, Doctoral Dissertation, University of California, Berkeley, 1972.Google Scholar
2. Buchwalter, H., Fonctions continues et mesures sur un espace complètement régulier, Springer- Verlag Lecture Notes 331 (1973), 183–202.Google Scholar
3. Dazord, J. and Jourlin, M., Sur les precompacts d'un espace localement convexe, C. R. Acad. Sci. Paris 274 (1972), 463-466, 550–553.Google Scholar
4. Dugundji, J., Topology (Allyn & Bacon, Boston, 1966).Google Scholar
5. Haydon, R., Compactness in CS(T) and applications, Publ. Dep. Math. Lyon 9 (1972), 105–113.Google Scholar
6. Haydon, R., On compactness in spaces of measures and measure-compact spaces, Proc. Lond. Math. Soc. 29 (1974), 1–16.Google Scholar
7. Haydon, R., Sur un problème de H. Buchwalter, C. R. Acad. Sci. Paris 275 (1972), 1077–1080.Google Scholar
8. Kannan, V. and Rajagopalan, M., On scattered spaces, Proc. Amer. Math. Soc. 43 (1974), 402–408.Google Scholar
9. Malliha Devi, K., Meyer, P., and Rajagopalan, M., When does countable compactness imply sequential compactness?, Gen. Top. and its Applications, to appear.Google Scholar
10. Noble, N., Countably compact and pseudocompact products, Czech. Math. J. 19 (94) (1969), 390–397.Google Scholar
11. Ptak, V., Compact subsets of convex topological linear spaces, Czech. Math. J. 4 (79) (1954), 51–74 (Russian. English summary).Google Scholar
12. Warner, S., The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265–282.Google Scholar