Linked bicontinuity of posets of compact ultrapseudometrics

Abstract

Approximation relations (``way below'' and ``way above'') onpointwise ordered sets of ultrapseudometrics are studied. It isproved that, to obtain continuity and/or dual continuity of theseposets in the sense considered in domain theory, i.e., possibilityto ``safely'' approximate all elements from below and from aboverespectively, one should restrict to compact ultrapseudometrics,not exceeding a fixed one. Such poset is a complete lattice,the partial order on it determines the Lawson topology and the dualLawson topology, which coincide (which is linked bicontinuity), arecompact Hausdorff, and agree with the uniform convergence metric.Necessary and sufficient conditions are proved for ``way below''and ``way above'' relations to hold. MSC Classification: 06F30 , 54E35