Quasi-Uniform Spaces in the Year 2001

Künzi, Hans-Peter A.

doi:10.1016/b978-044450980-2/50012-1

Cited by 8 publications

(4 citation statements)

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“…These spaces were first studied in relation with topological groups (see [11] for a recent survey). They have also become a center of interest for computer science since the pioneering works of Smyth and Sunderhauf [16], [17], [18], [19], [20], [21].…”

Section: Introductionmentioning

confidence: 99%

Enriched Categories and Quasi-uniform Spaces

Schmitt¹

2004

Electronic Notes in Theoretical Computer Science

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The metric space representation in terms of enriched categories of [14] is extended to quasi-uniform spaces and uniformly continuous maps. The bicompletion of the quasi-uniform spaces [6] is related to the Cauchy-completion. On one hand abstract quasi-uniformities and uniformly continuous maps are defined. They generalize quasi-uniform spaces and their morphisms and form a locally preordered 2-category AQU nif . On the other hand a 2-category Enr(H) of enrichments over different bases is defined. H stands here for a parameter 2-functor with domain a locally filtered 2-category and codomain the 2-category with objects monoidal partial orders and arrows the socalled super monoidal functors. Also the bases considered are partial orders (or "quantales"). In Enr(H) the Cauchy-complete objects defines a fully reflective sub-2-category. Eventually we show a 2-equivalence AQU nif ∼ = Enr(H) for a particular H. Thus one gets a completion for abstract quasi-uniformities, that reduces to the bicompletion for quasi-uniformities.

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Section: Introductionmentioning

confidence: 99%

Enriched Categories and Quasi-uniform Spaces

Schmitt¹

2004

Electronic Notes in Theoretical Computer Science

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“…An account of the theory of quasi-metric and quasi-uniform spaces up to 1982 is given in the book by Fletcher and Lindgren [9]. The survey papers by Künzi [18][19][20][21][22] are good guides for subsequent developments. Another book on quasi-uniform spaces is [26].…”

Section: Introductionmentioning

confidence: 99%

Compact and precompact sets in asymmetric locally convex spaces

Cobzaş¹

2009

Topology and its Applications

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MSC:primary 46A03 secondary 46A50, 46B07, 54E15, 54E25, 54E99 Keywords:Quasi-metric spaces Quasi-uniform spaces Spaces with asymmetric norm Asymmetric locally convex spaces Compactness Precompactness Total boundednessThe aim of the present paper is to study precompactness and compactness within the framework of asymmetric locally convex spaces, defined and studied by the author in [S. Cobza¸s, Asymmetric locally convex spaces, Int. J. Math. Math. Sci. 2005 (16) (2005) 2585-2608]. The obtained results extend some results on compactness in asymmetric normed spaces proved by [L.M. García-Raffi, Compactness and finite dimension in asymmetric normed linear spaces, Topology Appl. 153 (2005) 844-853], and [C. Alegre, I. Ferrando, L.M. García-Raffi, E.A. Sánchez-Pérez, Compactness in asymmetric normed spaces, Topology Appl. 155 (6) (2008) 527-539].

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“…It is well-known that (quasi-)uniformity is a very important concept close to topology and a convenient tool for investigating topology (see [4,13,15]). L-(quasi-)uniformity in Hutton's sense (see [9]) has been accepted by many authors and has attracted wide attention in the literature.…”

Section: Introductionmentioning

confidence: 99%