Nonstandard methods of completing quasi-uniform spaces

Render, Hermann

doi:10.1016/0166-8641(94)00052-5

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…The situation is different when considering quasimetric spaces and quasi-uniform spaces. The lack of symmetry (see [8] for a detailed account of symmetry and completions in the context of quantaloid enriched categories) sabotages the standard completion constructions that work in the symmetric case and the theory bifurcates with several different notions of complete objects and different completion processes existing in the literature (see e.g., [1,[3][4][5][6][9][10][11][12][13][14][15]).…”

Section: Introductionmentioning

confidence: 99%

Completion of continuity spaces with uniformly vanishing asymmetry

Chand

Weiss

2015

Topology and its Applications

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The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in question. For qausi-metric spaces and quasi-uniform spaces various non-equivalent completions exist, often defined on a certain subcategory of spaces that satisfy a key property required for the particular completion to exist. The classical filter completion of a uniform space can be adapted to yield a filter completion of a metric space. We show that this completion by filters generalizes to continuity spaces that satisfy a form of symmetry which we call uniformly vanishing asymmetry.

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Section: Introductionmentioning

confidence: 99%

Completion of continuity spaces with uniformly vanishing asymmetry

Chand

Weiss

2015

Topology and its Applications

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“…As an illustration, let us consider the following two very similar assertions: (i) ( [16] 6.4) a uniformly regular extension of a totally bounded quasiuniformity is totally bounded; (ii) ( [14] Lemma 1) a uniformly regular extension of a Cauchy bounded quasi-uniformity is Canchy bounded. [] 2.3.…”

Section: Cll(5)uoanx) C5)ua (Acy)mentioning

confidence: 99%

Short notes on quasi-uniform spaces II. Doubly uniformly strict extensions

Deak

1995

Acta Math Hung

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Uniformly strict extensions were introduced in [1], called there "strict extensions" (see w for the definition). It is the aim of this note to give a complete description of the doubly uniformly strict extensions (cf.[6] Problem 58A): V is called a doubly uniformly strict extension of/4 provided that 1; is a uniformly strict extension of/4, and V -1 of U -1. We shall also consider the smaller class of doubly uniformly regular extensions (see w for the definition). Formally, there is only a slight difference between the two results: the filters figuring in the first one are replaced by the associated grills in the second one.w Uniformly strict extensions 0.1. Throughout this paper X will denote a non-empty set, U a quasiuniformity on X and Y D X. A quasi-uniformity l/on Y is an extension of b/if Y]X =/4 and x is Tv-dense in Y; 13 is a double extension if, in addition, X is Tv-l-dense in Y. (The terminology was different in [4] to [9].) For a E C Y, [l(a) denotes the trace on X of the Tv-neighbourhood filter of a, called the trace filter of a; in the case of double extensions, [-l(a) denotes the trace on X of the Tv-l-neighbourhood filter of a, and (f-l(a), ~l(a)) is called the trace filter pair of a. In other words:

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“…Unfortunately, although too many current examples of quasi-uniform spaces have D-completion, that was not a resolution: "the quiet spaces constitute a very small class of spaces"(cf. [12]). Fletcher, in one his reviewing, notes: "D.Doitchinov showed that it is impossible to give a satisfactory theory of completion for the class of quasi-uniform spaces and introduced the class of quiet quasi-uniform spaces, a class comprising all uniform spaces for which a satisfactory theory of completion exists".…”

Section: Introductionmentioning

confidence: 99%

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

Andrikopoulos

2013

International Journal of Mathematics and Mathematical Sciences

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The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given 0 quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.

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Nonstandard methods of completing quasi-uniform spaces

Cited by 6 publications

References 15 publications

Completion of continuity spaces with uniformly vanishing asymmetry

Completion of continuity spaces with uniformly vanishing asymmetry

Short notes on quasi-uniform spaces II. Doubly uniformly strict extensions

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

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