Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T08:36:59.420Z Has data issue: false hasContentIssue false

Topological properties of the scale of a uniform space

Published online by Cambridge University Press:  09 April 2009

G. D. Richardson
Affiliation:
Department of Mathematics East Carolina University Greenville, North Carolina 27834, U.S.A.
E. M. Wolf
Affiliation:
Department of Mathematics Marshall University Huntington, West Virginia 25701, U.S.A.
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.

Let (S. U) be a uniform space. This space can be embedded in a complete, uniform lattice called the scale of (S. U). We prove that the scale is compact if and only if S is finite or U = {S × S}. We prove that this statement remains true if compact is replaced by countably compact, totally bounded. Lindelof, second countable, or separable. In the last section of this paper, we investigate the cardinality of the scale and the retracted scale.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Bourbaki, N., Topologie générale, Chapt. I, II (Paris, 1951).Google Scholar
[2]Bushaw, D., ‘A stability criterion for general systems’, Math. Svsteois Theory 1 (1967), 7988.CrossRefGoogle Scholar
[3]Kent, D. C., ‘On the scale of a uniform space’, Invent. Math. 4 (1967), 159164.CrossRefGoogle Scholar
[4]Leslie, G. S. and Kent, D. C., ‘Connectedness in the scale of a uniform space’, J. Austral. Math. Soc. 13 (1972), 305312.CrossRefGoogle Scholar
[5]Richardson, G. D., ‘Connectedness of uniform subspaces of R’. J. Austral. Math. Soc. 18 (1974), 461463.CrossRefGoogle Scholar