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Nowhere Dense Subsets of Metric Spaces with Applications to Stone-Cech Compactifications

Published online by Cambridge University Press:  20 November 2018

Jack R. Porter  and
R. Grant Woods
Jack R. Porter
Affiliation:
University of Kansas, Lawrence, Kansas
R. Grant Woods
Affiliation:
The University of Manitoba, Winnipeg, Manitoba
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Let X be a metric space. Assume either that X is locally compact or that X has no more than countably many isolated points. It is proved that if F is a nowhere dense subset of X, then it is regularly nowhere dense (in the sense of Katětov) and hence is contained in the topological boundary of some regular-closed subset of X. This result is used to obtain new properties of the remote points of the Stone-Čech compactification of a metric space without isolated points.

Let βX denote the Stone-Čech compactification of the completely regular Hausdorff space X. Fine and Gillman [3] define a point p of βX to be remote if p is not in the βX-closure of a discrete subset of X.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 622 - 630
Copyright
Copyright © Canadian Mathematical Society 1972

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