Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-21T23:54:31.520Z Has data issue: false hasContentIssue false
  • English
  • Français

The metrisability of precompact sets

Published online by Cambridge University Press:  17 April 2009

Neill Robertson
Neill Robertson
Affiliation:
Department of Mathematics, University of Cape Town, Rondebosch 7700, South Africa
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.

A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 131 - 135
Copyright
Copyright © Australian Mathematical Society 1991

References

[l]Cascales, B. and Orihuela, J., ‘On compactness in locally convex spaces’, Math. Z. 195 (1987), 365–381.CrossRefGoogle Scholar
[2]Drewnowski, L., ‘Another note on Kalton's Theorem’, Studia Math. 52 (1975), 233–237.CrossRefGoogle Scholar
[3]Hager, A.W., ‘Some nearly fine uniform spaces’, Proc. Lond. Math. Soc. (3) 28 (1974), 517–546.CrossRefGoogle Scholar
[4]Isbell, J.R., Uniform spaces (Amer. Math. Soc., Providence, 1964).CrossRefGoogle Scholar
[5]Jarchow, H., Locally convex spaces (Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[6]Pfister, H., ‘Bemerkungen zum Satz über die separabilität der Fréchet-Montel-Räume’, Arch. Math. 27 (1976), 86–92.CrossRefGoogle Scholar
[7]Schmets, J., Espaces de fonctions continues: Lecture Notes in Mathematics 519 (Springer-Ver Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar