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When exactly is Scott sober?
Citation
Ho, W. K., & Zhao, D. (2010). When exactly is Scott sober? (Report No. M2010-02). National Institute of Education (Singapore). NIE Digital Repository. https://hdl.handle.net/10497/15606
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
Publisher
National Institute of Education (Singapore)
Description
Technical report M2010-02, September 2010, Mathematics and Mathematics Education, National Institute of Education, Singapore