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The Duality of Distributive Continuous Lattices

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski
B. Banaschewski*
Affiliation:
McMaster University, Hamilton, Ontario
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Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.

Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 385 - 394
Copyright
Copyright © Canadian Mathematical Society 1980

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