## Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding (1996)

Citations: | 23 - 3 self |

### BibTeX

@MISC{Bonsangue96generalizedmetric,

author = {M.M. Bonsangue and F. van Breugel and J. J. M. M. Rutten},

title = {Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enriched-categorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the ffl-ball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.

### Citations

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(Show Context)
Citation Context ...ctions that it gives rise to elegant definitions and characterizations of completion, topology, and powerdomains. A general proof of the Yoneda Lemma for arbitrary enriched categories can be found in =-=[Kel82]-=-. For generalized metric spaces, it is proved in [Law86]. The proof is repeated here. The following notation will be used throughout the rest of this paper:sX = [0; 1] X op ; i.e., the set of all non-... |

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Citation Context ...both preordered spaces and ordinary metric spaces, which is the main motivation for the present study. Our sources of inspiration are the work of Lawvere on V-categories and generalized metric spaces =-=[Law73]-=- and the work by Smyth on quasi metric spaces [Smy91], and we have been influenced by recent work of Flagg and Kopperman [FK] and Wagner [Wag94]. The present paper continues earlier work [Rut96a], in ... |

52 |
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Citation Context ...limm bm ]: 2 5 Completion via Yoneda The completion of gms's is defined by means of the Yoneda embedding. It yields for ordinary metric spaces Hausdorff's standard Cauchy completion (as introduced in =-=[Hau14]-=-), for preorders the chain completion, and for qms's a completion given by Smyth (see [Smy91, page 214]). Let X be a gms. Because [0; 1] is a complete qms (cf. Section 2 and 3), it follows from Propos... |

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(Show Context)
Citation Context ...paces [Law73] and the work by Smyth on quasi metric spaces [Smy91], and we have been influenced by recent work of Flagg and Kopperman [FK] and Wagner [Wag94]. The present paper continues earlier work =-=[Rut96a]-=-, in which part of the theory of generalized metric spaces has been developed. The guiding principle throughout is Lawvere's view of metric spaces as [0; 1]-categories , by which they are structures t... |

34 |
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Citation Context ...re categories has been the main motivation for Lawvere in his study of generalized metric spaces as enriched categories [Law73]. Lawvere's work together with the more topological perspective of Smyth =-=[Smy88]-=- have been our main source of inspiration for the present paper which continues the work of Rutten [Rut96a]. Generalized metric spaces are a special instance of Lawvere's V-categories. The non-symmetr... |

27 |
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(Show Context)
Citation Context ... metric domain theory. Both disciplines play a central role in (to a large extent even came into existence because of) the semantics of programming languages (cf. recent textbooks such as [Win93] and =-=[BV96]-=-, respectively). The use of generalized metric spaces in semantics, or more precisely, in the study of transition systems, will be an important next step. The combination of results from [Rut96a] (on ... |

26 | General Topology, volume 6 - Engelking - 1989 |

20 |
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Citation Context ...n are the work of Lawvere on V-categories and generalized metric spaces [Law73] and the work by Smyth on quasi metric spaces [Smy91], and we have been influenced by recent work of Flagg and Kopperman =-=[FK]-=- and Wagner [Wag94]. The present paper continues earlier work [Rut96a], in which part of the theory of generalized metric spaces has been developed. The guiding principle throughout is Lawvere's view ... |

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15 | Generalized ultrametric spaces: completion, topology, and powerdomains via the Yoneda embedding
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(Show Context)
Citation Context ...e one hand the familiar lower, upper, and convex powerdomains from order-theory; and on the other hand the metric powerdomain of compact subsets. The present paper is a reworking of an earlier report =-=[BBR95]-=-, in which generalized ultrametric spaces are considered, satisfying X(x; z)smaxfX(x; y); X(y; z)g, for all x, y, and z in X . There is but little difference between the two papers: as it turns out, n... |

15 |
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Citation Context ...r that name, in [JMP86, Lemma II-2.8]. The comprehension schema as a comparison between predicates and subsets has been studied in the context of generalized metric spaces by Lawvere [Law73] and Kent =-=[Ken88]-=-. The definition of the generalized Scott topology via the Yoneda embedding seems to be new while the direct definition---by specifying the open sets---is briefly mentioned in the conclusion of [Smy88... |

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8 |
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Citation Context ...definition of a completion of monoidal closed categories. The use of the Yoneda lemma for the completion of generalized metric spaces is new, but it is suggested by an embedding theorem of Kuratowski =-=[Kur35]-=- and the definition of completion as in [Eng89, Theorems 4.3.13-4.3.19] for standard metric spaces. A metric version of the Yoneda lemma also occurs, though not under that name, in [JMP86, Lemma II-2.... |

8 |
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(Show Context)
Citation Context ...se to completion is well known for many mathematical structures such as groups, lattices, and categories. In [Wag95], an enriched version of the Dedekind-MacNeille completion of lattices is given. In =-=[Sas94]-=-, the Yoneda lemma is used in the definition of a completion of monoidal closed categories. The use of the Yoneda lemma for the completion of generalized metric spaces is new, but it is suggested by a... |

5 | Topological spaces for cpos - Melton - 1989 |

5 |
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- Rutten
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(Show Context)
Citation Context ...etric space is from [Smy88] as well as the notion of limit. A purely enriched-categorical definition of forward Cauchy sequences and of limits can be found in Wagner's [Wag94, Wag95]. In [Rut96a] and =-=[Rut96b]-=-, the definitions of forward limit and backward limit are shown to be special instances of the enriched-categorical notions of weighted limit and weighted colimit. The notions of finiteness and algebr... |

4 | Alexandroff and Scott topologies for generalized metric spaces
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- 1996
(Show Context)
Citation Context ...or all x, y, and z in X . There is but little difference between the two papers: as it turns out, none of the proofs about ultrametrics relies essentially on the strong triangle inequality. (See also =-=[BBR]-=-, which contains part of the present paper.) As mentioned above, generalized metric spaces and the constructions that are given in the present paper both unify and generalize a substantial part of ord... |

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3 | The essence of ideal completion in quantitative form
- Flagg, Sunderhauf
- 1996
(Show Context)
Citation Context ...the completion of X . An interesting question is to characterize the family of gms's for which completion is idempotent. Clearly it contains all ordinary metric spaces. Recently, Flagg and Sunderhauf =-=[FS96]-=- have answered this question: the completion of the completion of a gms X is isomorphic to the completion of X if and only if the generalized Alexandroff and the generalized Scott topologies on X (to ... |

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