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When exactly is Scott sober?
Abstract
A topological space is sober if every nonempty irreducible closed set is the closure of a unique singleton set. Sobriety is precisely the
topological property that allows one to recover completely a topological space from its frame of opens. Because every Hausdor space is
sober, sobriety is an overt, and hence unnamed, notion. Even in non-Hausdor settings, sober spaces abound. A well-known instance of a
sober space appears in domain theory: the Scott topology of a continuous dcpo is sober. The converse is false as witnessed by two counterexamples constructed in the early 1980's: the first by P.T. Johnstone and
the second (a complete lattice) by J. Isbell. Since then, there has been
limited progress in the quest for an order-theoretic characterization of
those dcpo's for which their Scott topology is sober. This paper provides one answer to this open problem.
Date Issued
2010
Description
Technical report M2010-02, September 2010, Mathematics and Mathematics Education, National Institute of Education, Singapore