## Alexandroff and Scott Topologies for Generalized Metric Spaces (0)

Venue: | Proceedings of the 11th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Bonsangue_alexandroffand,

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

title = {Alexandroff and Scott Topologies for Generalized Metric Spaces},

booktitle = {Proceedings of the 11th Summer Conference on General Topology and Applications, Annals of the New York Academy of Sciences},

year = {},

pages = {49--68}

}

### OpenURL

### Abstract

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Every generalized metric space can be isometrically embedded in a complete function space by means of a metric version of the categorical Yoneda embedding. This simple fact gives naturally rise to: 1. a topology for generalized metric spaces which for arbitrary preorders corresponds to the Alexandroff topology and for ordinary metric spaces reduces to the ffl-ball topology; 2. a topology for algebraic generalized metric spaces generalizing both the Scott topology for algebraic complete partial orders and the ffl-ball topology for metric spaces. AMS subject classification (1991): 68Q10, 68Q55 Keywords: generalized metric, preorder, metric, Alexandroff topology, Scott topology, ffl-ball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been develop...

### Citations

393 | Basic concepts of enriched category theory - Kelly - 2005 |

244 |
The Formal Semantics of Programming Languages: an Introduction
- Winskel
- 1993
(Show Context)
Citation Context ...ic, Alexandroff topology, Scott topology, ffl-ball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., =-=[Win93]-=- and [BV96]). Parts of their theory have been developed because of semantic necessity (see, e.g., [SP82] and [AR89]). Generalized metric spaces provide a common framework for the study of both preorde... |

177 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
- 1982
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Citation Context ...rs and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been developed because of semantic necessity (see, e.g., =-=[SP82]-=- and [AR89]). Generalized metric spaces provide a common framework for the study of both preorders and ordinary metric spaces. A generalized metric space (gms for short) consists of a set X together w... |

116 |
Metric spaces, generalized logic and closed categories
- Lawvere
- 1974
(Show Context)
Citation Context ...cott topology in terms of a closure operator. Both closure operators are defined by means of an adjunction between preorders. In defining these adjunctions we use the fact---first observed by Lawvere =-=[Law73]-=----that, intuitively, one may identify elements x of a gms X with a description of the distances between any element y in X and x. Formally, this description is a function mapping every y in X to the ... |

72 |
Solving reflexive domain equations in a category of complete metric spaces
- America, Rutten
- 1989
(Show Context)
Citation Context ...ic spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and [BV96]). Parts of their theory have been developed because of semantic necessity (see, e.g., [SP82] and =-=[AR89]-=-). Generalized metric spaces provide a common framework for the study of both preorders and ordinary metric spaces. A generalized metric space (gms for short) consists of a set X together with a dista... |

37 |
Elements of generalized ultrametric domain theory
- Rutten
- 1996
(Show Context)
Citation Context ...e enriched categorical approach of Lawvere [Law73, Law86] (cf. [Ken87, Wag94, Wag95]) and the topological approach of Smyth [Smy87, Smy91] (cf. [FK94]). The present paper continues the work of Rutten =-=[Rut95]-=- and is part of [BBR95]. In the latter paper, besides the topologies presented in this paper also completion and powerdomains for generalized ultrametric spaces are defined by means of the Yoneda embe... |

36 |
Totally bounded spaces and compact ordered spaces as domains of computation
- Smyth
- 1991
(Show Context)
Citation Context ...ning the generalized Scott topology. Like the ordinary Scott topology for complete partial orders, the generalized Scott topology encodes all information about order, convergence, and continuity (cf. =-=[Smy91]-=-). The generalized Alexandroff topology only encodes the information about order, just like the ordinary Alexandroff topology for preorders (cf. [Smy87, FK94]). The paper is organized as follows. Sect... |

35 |
Quasi-uniformities: Reconciling domains with metric spaces
- Smyth
- 1988
(Show Context)
Citation Context ...s similar to the ones given by Smyth [Smy87, Smy91] and Flagg and Kopperman [FK94]. A definition of a generalized Scott topology in terms of open sets similar to ours is briefly mentioned by Smyth in =-=[Smy87]-=-. The definition of the generalized Alexandroff topology in terms of a closure operator already appears in [Law73, Law86, Ken87]. New is the definition of the generalized Scott topology in terms of a ... |

27 |
Control Flow Semantics. Foundations of Computing Series
- Bakker, Vink
- 1996
(Show Context)
Citation Context ...off topology, Scott topology, ffl-ball topology, Yoneda embedding 1 Introduction Partial orders and metric spaces play a central role in the semantics of programming languages (see, e.g., [Win93] and =-=[BV96]-=-). Parts of their theory have been developed because of semantic necessity (see, e.g., [SP82] and [AR89]). Generalized metric spaces provide a common framework for the study of both preorders and ordi... |

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

21 | Solving Recursive Domain Equations with Enriched Categories - Wagner - 1994 |

20 |
Continuity spaces: reconciling domains and metric spaces
- Flagg, Kopperman
- 1995
(Show Context)
Citation Context ...lternative definitions are shown to coincide. Our definition of the generalized Alexandroff topology in terms of open sets is similar to the ones given by Smyth [Smy87, Smy91] and Flagg and Kopperman =-=[FK94]-=-. A definition of a generalized Scott topology in terms of open sets similar to ours is briefly mentioned by Smyth in [Smy87]. The definition of the generalized Alexandroff topology in terms of a clos... |

15 | Generalized ultrametric spaces: completion, topology, and powerdomains via the Yoneda embedding
- Breugel, Rutten
- 1995
(Show Context)
Citation Context ...g y isometrically embeds a gms X into the complete gms b X. One can define the completion of X as the smallest complete subspace of b X containing the y-image of X. For details we refer the reader to =-=[BBR95]-=-. 7 The preordered notion finite can be generalized as follows. An element x in a gms X is finite if the function y 2 X:X (x; y) from X to [0; 1] is continuous. In order to conclude that x is finite i... |

15 | The metric closure powerspace construction - Kent - 1987 |

14 | Taking categories seriously - Lawvere - 2005 |

10 |
On the homology theory of modules
- Yoneda
- 1954
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Citation Context ...unction OE assigns to an element y in X is thought of as a measure for the extent to which y is an element of OE. This fact corresponds to a generalized metric version of the categorical Yoneda lemma =-=[Yon54]-=-. The corresponding Yoneda embedding isometrically embeds a gms X into the gms of fuzzy subsets of X . By comparing the fuzzy subsets of X with the ordinary subsets of X we obtain an adjunction. This ... |

5 | Topological spaces for cpos - Melton - 1989 |

4 |
Liminf convergence in !-categories
- Wagner
- 1995
(Show Context)
Citation Context ...theorems for ordinary metric spaces can be adapted. The price to be paid for this simplicity is that most of their results only hold for a restricted class of spaces: they have to be spectral. Wagner =-=[Wag95]-=- has also presented a generalized Scott topology. Although for complete partial orders his topology corresponds to the Scott topology, for metric spaces it does not coincide with the ffl-ball topology... |

3 | The essence of ideal completion in quantitative form - Flagg, Sunderhauf - 1996 |