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Domain Representations of Topological Spaces
, 2000
"... A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, ..."
Abstract

Cited by 25 (9 self)
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A domain representation of a topological space X is a function, usually a quotient map, from a subset of a domain onto X . Several different classes of domain representations are introduced and studied. It is investigated when it is possible to build domain representations from existing ones. It is, for example, discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors. It is shown that any T 0 topological space has a domain representation. These domain representations are very large. However, smaller domain representations are also constructed for large classes of spaces. For example, each second countable regular Hausdorff space has a domain representation with a countable base. Domain representations of functions and function spaces are also studied.
Effective Domain Representations of H(X), the space of compact subsets
, 1999
"... This paper gives effective domain representations of spaces H(X) of nonempty compact subsets of effective complete metric spaces X. The domain representation of H(X) is constructed from a domain representation of X using the Plotkin power domain construction. As an application of the representation ..."
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Cited by 13 (5 self)
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This paper gives effective domain representations of spaces H(X) of nonempty compact subsets of effective complete metric spaces X. The domain representation of H(X) is constructed from a domain representation of X using the Plotkin power domain construction. As an application of the representation an effective version of a fundamental theorem on IFS (iterated function system) is shown.
On the Equivalence of Some Approaches to Computability on the Real Line
"... There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch's Type Two Theory of Effectivity or on domaintheoretic approaches, in which case the part ..."
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Cited by 3 (0 self)
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There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch's Type Two Theory of Effectivity or on domaintheoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysisoriented line of research is based on Grzegorczyk's work. All these approaches are claimed to be equivalent, but not in all cases proofs have been given. In this paper it is shown that a real number as well as a realvalued function are computable in Weihrauch's sense if and only if they are definable in Escardo's functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this doma...
Streams, Stream Transformers and Domain Representations
 Prospects for Hardware Foundations, Lecture Notes in Computer Science
, 1998
"... We present a general theory for the computation of stream transformers of the form F: (R B) (T A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains can be applied to transformations of continuous ..."
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Cited by 3 (3 self)
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We present a general theory for the computation of stream transformers of the form F: (R B) (T A), where time T and R, and data A and B, are discrete or continuous. We show how methods for representing topological algebras by algebraic domains can be applied to transformations of continuous streams. A stream transformer is continuous in the compactopen topology on continuous streams if and only if it has a continuous lifting to a standard algebraic domain representation of such streams. We also examine the important problem of representing discontinuous streams, such as signals T A, where time T is continuous and data A is discrete.
Fundamentals of Computing I
 Logic, Problem Solving, Programs, & Computers
, 1992
"... on topological spaces via domain representations ..."