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Cocompactness in algebra and topology

Part of: Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Graham J. Logan
Graham J. Logan
Affiliation:
Department of Applied Sciences Christchurch Polytechnic, New Zealand
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Abstract

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This paper discusses two notions, developed independently and both termed “cocompactness”. The first arises in the area of topology, where J. de Groot and others have studied spaces which are, in a certain sense, complementary to a given space. If the given space is compact then the complementary spaces are said to be cocompact. The second concept arises in the area of logic and general algebra. Loosely speaking a logic is compact if every inconsistent set of formulas has a finite inconsistent subset. This notion of compactness may be generalized to any closure algebra and the use of the term “cocompactness” to describe the generalization was suggested to the author by Dr. R. A. Bull.

It is shown here that topological and algebraic cocompactness are related in the following ways. Firstly, if a closure algebra is algebraically cocompact then its dual space is topologjcally cocompact, and conditions may be given for the implication to be reversible.3 Furthermore any cocompact topological space may be represented as the continuous 1-1 image of the dual space of a cocompact closure algebra. A final result relates another class of closure algebras with those topological spaces which are compact.

MSC classification

Secondary: 54D30: Compactness
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 4 , December 1981 , pp. 385 - 389
Copyright
Copyright © Australian Mathematical Society 1981

References

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