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Regular Wallman compactifications of rim-compact spaces

Part of: Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Olav Njåstad
Olav Njåstad
Affiliation:
Department of Mathematics, University of Trondheim-Norwegian, Institute of TechnologyN-7034 Trondheim, Norway
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Abstract

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A compact Hausdorff space is regular Wallman if it possesses a separating ring of regular closed sets, an s-ring. It was proved by P. C. Baayen and J. van Mill [General Topology and Appl. 9 (1978), 125–129] that if a locally compact Hausdorff space possesses an s-ring, then every Hausdorff compactification with zero-dimensional remainder is regular Wallman.

In this paper the reasoning leading to this result is modified to work in a more general setting. Iet αX be a Hausdorff compactification of a space X, and let be the family of those closed sets in αX whose boundaries are contained in X. A main result is the following: If contains an s-ring for some Hausdorff compactification γX, then every larger Hausdorff compactification αX for which is a base for the closed sets on αX — X, is regular Wallman. Various consequences concerning compactifications of a class of rim-compact spaces (called totally rim-compact spaces) are discussed.

MSC classification

Secondary: 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 3 , October 1981 , pp. 312 - 324
Copyright
Copyright © Australian Mathematical Society 1981

References

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