Please use this identifier to cite or link to this item: http://hdl.handle.net/10497/15606
Title: 
When exactly is Scott sober?
Authors: 
Keywords: 
Scott topology
Sober space
dcpo
Dominated dcpo,
H-continuous
H-algebraic
H-compact
Strongly H-algebraic
Issue Date: 
2010
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.
Description: 
Technical report M2010-02, September 2010, Mathematics and Mathematics Education, National Institute of Education, Singapore
URI: 
Appears in Collections:Research Reports

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