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On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom

Part of: Ordered sets Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Karl H. Hofmann  and
Jimmie D. Lawson
Karl H. Hofmann
Affiliation:
Department of Mathematics Tulane UniversityNew Orleans, Louisiana 70018 U.S.A. Fachbereich Mathematik THD D 6100 Darmstadt FR, Germany
Jimmie D. Lawson
Affiliation:
Department of Mathematics Louisiana State UniversityBaton Rouge, Louisiana 70803, U.S.A.
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Abstract

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A Hausdorff space X is said to be compactly generated (a k-space) if and only if the open subsets U of X are precisely those subsets for which K ∩ U is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice O(X) of open sets of X. This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces.

Keywords

k-space; compactly generated spacedualitysemilatticecontinuous lattice.

MSC classification

Secondary: 06A12: Semilattices 06A15: Galois correspondences, closure operators 54D50: $k$-spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 2 , April 1984 , pp. 194 - 212
Copyright
Copyright © Australian Mathematical Society 1984

References

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