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Topological properties of the scale of a uniform space

Part of: Spaces with richer structures Fairly general properties

Published online by Cambridge University Press:  09 April 2009

G. D. Richardson  and
E. M. Wolf
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.
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Abstract

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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.

Keywords

uniform spacescaleretracted scalecompacttotally boundedseparablecardinality.

MSC classification

Secondary: 54D30: Compactness 54D45: Local compactness, $sigma$-compactness 54E15: Uniform structures and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 54 - 61
Copyright
Copyright © Australian Mathematical Society 1982

References

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