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Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' ..."
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Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
Polish spaces, computable approximations, and bitopological spaces
, 2000
"... Answering a question of J. Lawson (formulated also earlier, in 1984, by Kamimura and Tang [16]) we show that every Polish space admits a bounded complete computational model, as defined below. This results from our construction, in each Polish space 〈X, τ〉, of a countable family C of nonempty close ..."
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Answering a question of J. Lawson (formulated also earlier, in 1984, by Kamimura and Tang [16]) we show that every Polish space admits a bounded complete computational model, as defined below. This results from our construction, in each Polish space 〈X, τ〉, of a countable family C of nonempty closed subsets of X such that: (cp) each subset of C with the finite intersection property has nonempty intersection;
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"... Abstract In this article we show that each complete metric space is the maximal point space of a continuous, bounded complete dcpo (in other common terminology, we show that such a space has a bounded complete computational model). This gives a positive answer to J. Lawson's question of whether ..."
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Abstract In this article we show that each complete metric space is the maximal point space of a continuous, bounded complete dcpo (in other common terminology, we show that such a space has a bounded complete computational model). This gives a positive answer to J. Lawson's question of whether each completely metrizable space is the maximal point space of a continuous, bounded complete dcpo. The proposed solution to this wellknown problem exhibits some fundamental links that hold between bitopological spaces, asymmetric topological structures and bounded complete computational models. In fact, the existence of a computational model for a topological space turns out to be equivalent to a number of fairly familiar concepts from asymmetric topology.
Abstract How Do Domains Model Topologies?
"... In this brief study we explicitly match the properties of spaces modelled by domains with the structure of their models. We claim that each property of the modelled topology is coupled with some construct in the model. Examples are pairs: (i) firstcountability strictly monotone map, (ii) developabi ..."
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In this brief study we explicitly match the properties of spaces modelled by domains with the structure of their models. We claim that each property of the modelled topology is coupled with some construct in the model. Examples are pairs: (i) firstcountability strictly monotone map, (ii) developability measurement, (iii) metrizability partial metric, (iv) ultrametrizability tree, (v) Choquetcompletenessdcpo, and more. By making this correspondence precise and explicit we reveal how domains model topologies. 1