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A direct proof of the Hofmann-Mislove theorem. (English) Zbl 0789.54030

The theorem of the title is the assertion that, in the lattice of open subsets of a sober topological space \(X\), the filters which are Scott- open (i.e. inaccessible by directed joins) are exactly the neighbourhood filters of compact saturated subsets of \(X\). (The authors ascribe this result to a paper by K. H. Hofmann and M. W. Mislove published in 1981, although the reviewer has the impression that it had formed part of the “folklore” of the continuous-lattice community for some time before that.) The purpose of this note is to give what the authors claim to be a more direct proof of the theorem, although they freely acknowledge that the underlying idea is the same as in all previously-published proofs.

MSC:

54D30 Compactness
06B30 Topological lattices
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
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References:

[1] Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott, A compendium of continuous lattices, Springer-Verlag, Berlin-New York, 1980. · Zbl 0452.06001
[2] K. H. Hofmann and M. Mislove, Local compactness and continuous lattices, Continuous Lattices (Proceedings, Bremen, 1979) , Springer-Verlag, Berlin, 1981, pp. 209-248.
[3] Steven Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, vol. 5, Cambridge University Press, Cambridge, 1989. · Zbl 0668.54001
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