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QuasiUniform Completeness In Terms Of Cauchy Nets
 Acta Mathematica Hungarica
, 1993
"... . The notion of completeness for quasiuniform spaces developed in [Smy93] and [Sun93] is presented in terms of nets rather than filters. As a byproduct, we get a new characterisation of completability and the result that for all completable spaces the Smythcompletion coincides with the wellknown ..."
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. The notion of completeness for quasiuniform spaces developed in [Smy93] and [Sun93] is presented in terms of nets rather than filters. As a byproduct, we get a new characterisation of completability and the result that for all completable spaces the Smythcompletion coincides with the wellknown bicompletion of [FL82]. Totally bounded spaces are singled out to belong to the class of completable quasiuniform spaces. Introduction In [Smy93] and [Sun93], a theory of completion and completeness for quasiuniform spaces is developed. This paper presents completeness in terms of nets rather than filters. The usual way to pass from filters to nets by choosing arbitrary elements from the sets in the filter does not work in the nonsymmetric case of quasiuniformities. The reason for this is that our Cauchy filters do not contain sets which are arbitrarily small in an absolute sense, hence the resulting net need not be Cauchy. The completion of [Smy93] and [Sun93] is performed in a larger c...
On the Yoneda completion of a quasimetric space
 Theoretical Computer Science
, 2002
"... Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR ..."
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Cited by 8 (4 self)
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Several theories aimed at reconciling the partial order and the metric space approaches to Domain Theory have been presented in the literature (e.g. [FK97], [BvBR9 8], [Smy89] and [Wag94]). We focus in this paper on two of these approaches: the Yoneda completion of generalized metric spaces of [BvBR98], which finds its roots in work by Lawvere ([Law73], cf. also [Wag94]) and which is related to early work by Stoltenberg (e.g. [Sto67], [Sto67a] and [FG84]), and the Smyth completion ([Smy89],[Smy91],[Smy94],[Sun93] and [Sun95]). A netversion of the Yoneda completion, complementing the netversion of the Smyth completion ([Sun95]), is given and a comparison between the two types of completion is presented. The following open question is raised in [BvBR98]: "An interesting question is to characterize the family of generalized metric spaces for which [the Yoneda] completion is idempotent (it contains at least all ordinary metric spaces)." We show that the largest class of quasimetric spaces idempotent under the Yoneda completion is precisely the class of Smythcompletable spaces. A similar result has been obtained independently by B. Flagg and P. Sünderhauf in [FS96]
Quasimetric Properties of Complexity Spaces
 Topology Appl
, 1999
"... The complexity (quasimetric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms ([12]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [12]. Here we obtain sev ..."
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Cited by 7 (4 self)
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The complexity (quasimetric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms ([12]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [12]. Here we obtain several quasimetric properties of the complexity space. The main results obtained are the Smythcompleteness of the complexity space and the compactness of closed complexity spaces which possess a (complexity) lower bound. Finally, some implications of these results in connection to the above mentioned complexity analysis techniques are discussed and the total boundedness of complexity spaces with a lower bound is discussed in the light of Smyth's computational interpretation of this property ([14]). AMS (1991) Subject Classification: 54E15, 54E35, 54C30, 54C35. 1 Introduction The letters N , !, R and R + denote the set of positive integers, of nonnegative integers, of real numbers ...
The Essence of Ideal Completion in Quantitative Form
 GHK
, 1996
"... This paper is part of the ongoing foundational work on quantitative domain theory [Smy88,BvBR95,Rut96,FWS96,Sun94,Wag94], which refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of degree of approximation (cf. the introduction of [FWS96]). ..."
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Cited by 3 (0 self)
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This paper is part of the ongoing foundational work on quantitative domain theory [Smy88,BvBR95,Rut96,FWS96,Sun94,Wag94], which refines ordinary domain theory by replacing the qualitative notion of approximation by a quantitative notion of degree of approximation (cf. the introduction of [FWS96]). We investigate the generalization of ideal completion of posets for quantitative domains suggested in [BvBR95] and [FWS96].
Complexity Spaces Revisited (Extended Abstract)
"... The complexity (quasipseudometric) spaces have been introduced as part of the development of a topological foundation for the complexity analysis of algorithms ([Sch95]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [Sch95]. Typically ..."
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Cited by 3 (3 self)
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The complexity (quasipseudometric) spaces have been introduced as part of the development of a topological foundation for the complexity analysis of algorithms ([Sch95]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [Sch95]. Typically these applications involve fixed point arguments based on the Smyth completion. The notion of Scomplet...
Constructing a QuasiUniform Function Space
, 1993
"... This paper attacks the problem of constructing function spaces for a convenient class of quasiuniform spaces. As, for the sake of completeness, multivalued functions have to be considered, we define a suitable power space functor. The arising monad is a computational monad in the sense of Eugenio ..."
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This paper attacks the problem of constructing function spaces for a convenient class of quasiuniform spaces. As, for the sake of completeness, multivalued functions have to be considered, we define a suitable power space functor. The arising monad is a computational monad in the sense of Eugenio Moggi. The tensorial strength thus given enables us to lift the usual product to a symmetric tensor product on the Kleislicategory of the monad. It is now possible to define a function space constructor yielding a symmetric monoidal closed category. As an example, we give a convenient model for the real numbers in this category. Introduction Quasiuniformities  a common refinement of partially ordered sets and uniform spaces  serve as a useful notion to introduce the idea of quantity into domain theory. This paper attacks the problem of constructing function spaces for a convenient class of quasiuniform spaces. As objects of our category we choose complete totally bounded quasiunifo...
Function Spaces for Uniformly Locally Bounded QuasiUniform Spaces
, 1995
"... In a previous paper [Sun95b], we described a function space constructor for complete totally bounded quasiuniform spaces. The present paper generalizes this construction to the wider class of uniformly locally bounded spaces, which is contained in the class of all completable spaces. Based on this, ..."
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In a previous paper [Sun95b], we described a function space constructor for complete totally bounded quasiuniform spaces. The present paper generalizes this construction to the wider class of uniformly locally bounded spaces, which is contained in the class of all completable spaces. Based on this, a refined function space construction is introduced. The latter seems more appropriate for the uniform case and for certain applications in programming semantics. The constructions are wellbehaved with regard to uniform local boundedness and completability. Introduction This paper is part of the program [Smy88, Smy91, Sun95b, FK96] of developing a quantitative version of domain theory, that is to provide the theory of quasiuniformities or related structures with constructions and notions from (ordinary) domain theory [AJ94]. Specifically, we consider the question of supplying suitable function space constructors, a question which is also of interest in the general theory of quasiuniform...
The essence of ideal completion in quantitative form (Extended Abstract)
, 1995
"... Robert C. Flagg and Philipp Sunderhauf y University of Southern Maine fflagg,psunderg@usm.maine.edu December 12, 1995 Abstract If a posets lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal comple ..."
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Robert C. Flagg and Philipp Sunderhauf y University of Southern Maine fflagg,psunderg@usm.maine.edu December 12, 1995 Abstract If a posets lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with Scottcontinuous functions as morphisms. The same applies for the quantitative version of ideal completion suggested in the literature. As in the case of posets, it seems advantageous to consider a different topology with the completed spaces. We introduce Smyth completion as tool to automatically end up with the right topology after completing. 1 Introduction This paper is part of the ongoing foundational work on quantitative domain theory [Smy88, BBR95, Rut95, FW95, Wag94], which refines ordinary do Supported by the Deutsche Forschungsgemeinschaf...
Smyth Completion as Bicompletion
, 1997
"... We define a functor from topological quasiuniform spaces and continuous, uniformly continuous maps to quasiuniform spaces and uniformly continuous maps. This functor retains the same underlying Supported by the Deutsche Forschungsgemeinschaft. y Research carried out during a visit to the Depa ..."
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We define a functor from topological quasiuniform spaces and continuous, uniformly continuous maps to quasiuniform spaces and uniformly continuous maps. This functor retains the same underlying Supported by the Deutsche Forschungsgemeinschaft. y Research carried out during a visit to the Department of Mathematics and Statistics, University of Southern Maine. 1991 Mathematics Subject Classification. Primary 54E15. Key words and phrases. Wellmonotone quasiuniformity, Smythcompletion, bicompletion. 1 set, and generalizes a construction due to Junnila [Jun78] and Kunzi and Ferrario [KF91] to obtain another quasiuniformity, with respect to which the continuous uniformly continuous functions on the original structure, are uniformly continuous; the functor then takes the maps to themselves. As a result, this functor takes the Smyth completion of a topological quasiuniform space into the bicompletion of its image under the functor, and a topological quasiuniform space is ...
The Ideal Completion is Not Sequentially Adequate
"... It is well known that for the case of a countable partial order, the ideal completion and the chain completion coincide. We investigate the boundary at which the chain and ideal completion do not coincide. We show in particular that the ideal completion is not sequentially adequate; that is it is ..."
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It is well known that for the case of a countable partial order, the ideal completion and the chain completion coincide. We investigate the boundary at which the chain and ideal completion do not coincide. We show in particular that the ideal completion is not sequentially adequate; that is it is not possible in general to simply replace the ideal completion with a completion based on sequences as for instance the chain completion. The implications of this result for the Yoneda completion ([BvBR98]) and for the Smyth completion ([Smy89],[Smy91],[Smy94],[Sun93] and [Sun95]) which are based on the ideal completion, are discussed in an extended version of this paper, reported in [KS98].