## Ideal Models of Spaces (2000)

Venue: | Theoretical Computer Science |

Citations: | 5 - 2 self |

### BibTeX

@ARTICLE{Martin00idealmodels,

author = {Keye Martin},

title = {Ideal Models of Spaces},

journal = {Theoretical Computer Science},

year = {2000}

}

### OpenURL

### Abstract

Ideal domains have an elementary order theoretic structure: Every element is either compact or maximal. Despite this, we establish that (1) They can model any space currently known to possess a countably based model, and (2) The metric spaces with ideal models are exactly the completely metrizable spaces.

### Citations

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Citation Context ...emma for ideal models of metric spaces in general, which enables the aforementioned domain theoretic account of complete metrization. 2 Background 2.1 Domain theory A poset is a partially ordered set =-=-=-[2]. Denition 2.1 Let (P; v) be a partially ordered set. A nonempty subset S P is directed if (8x; y 2 S)(9z 2 S) x; y v z. The supremum of a subset S P is the least of all its upper bounds provided... |

370 | Classical Descriptive Set Theory - Kechris - 1994 |

70 | Dynamical systems, measures and fractals via domain theory
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Citation Context ...odels are exactly the completely metrizable spaces. 1 Introduction Over ten years before the realization that certain parts of mathematics could be executed in a purely domain theoretic manner ([6][7]=-=[8]-=-[9]), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets ... |

60 | Domain theory and integration
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(Show Context)
Citation Context ...l models are exactly the completely metrizable spaces. 1 Introduction Over ten years before the realization that certain parts of mathematics could be executed in a purely domain theoretic manner ([6]=-=[7]-=-[8][9]), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one ge... |

48 |
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Citation Context ...nterval domain and it is a model of the real line since max IR = f[x] : x 2 Rg ' R: Example 2.13 A model for locally compact Hausdor spaces. If X is a locally compact Hausdor space, its upper space [14] UX = f; 6= K X : K is compactg ordered under reverse inclusion A v B , B A is a continuous dcpo. The supremum of a directed set S UX is T S and the approximation relation is A B , B int(A).... |

45 | A computational model for metric spaces
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(Show Context)
Citation Context ...deal models are exactly the completely metrizable spaces. 1 Introduction Over ten years before the realization that certain parts of mathematics could be executed in a purely domain theoretic manner (=-=[6]-=-[7][8][9]), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one... |

40 | A Foundation for Computation
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- 2000
(Show Context)
Citation Context ...g a one point space. Recently, though, a model of a one point space, namely N 1 = N [ f1g, ordered by x v y , (x; y 2 N & x y) or y = 1; was used to capture the class of partial recursive functions [=-=2-=-0]. The fact that maxN 1 is a one point space captures the intuition that innite loops correspond to meaningless computations. The same is not true of our next model, however, where an innite loop mig... |

34 |
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(Show Context)
Citation Context ...nuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing (=-=[17]-=-[18][11][5][12][23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inh... |

28 | The measurement process in domain theory
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- 2000
(Show Context)
Citation Context ...s the intuition that innite loops correspond to meaningless computations. The same is not true of our next model, however, where an innite loop might represent a process like a zerosnding algorithm [2=-=5]-=-, one which is capable of approximating something up to very high levels of accuracy. Example 2.12 A model of the real line. The collection of compact intervals of the real line IR = f[a; b] : a; b 2 ... |

16 | When Scott is weak at the top
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(Show Context)
Citation Context ...ls are exactly the completely metrizable spaces. 1 Introduction Over ten years before the realization that certain parts of mathematics could be executed in a purely domain theoretic manner ([6][7][8]=-=[9]-=-), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the... |

14 |
Measure theoretic results for continuous valuations on partially ordered spaces
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Citation Context ...ce by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23][24][26][27]=-=[19]-=-[28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. With emphasis: (a) T... |

11 |
A Glimm{Eros dichotomy for Borel equivalence relations
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Citation Context ...ausdor space is Choquet complete. (iii) A metric space is Choquet complete i it is completely metrizable. (iv) A Gssubset of a Choquet complete space is Choquet complete. A proof of (iv) appears in [1=-=3]-=-, and seems to be a matter of folklore. The others are due to Choquet [4]. Theorem 6.3 (Martin [27]) The Scott topology on a domain is Choquet complete. The proof of Theorem 2.15 works as follows: If ... |

9 |
Computational models for ultrametric spaces
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(Show Context)
Citation Context ...po, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18]=-=[11]-=-[5][12][23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited f... |

9 | Non-classical techniques for models of computation
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(Show Context)
Citation Context ...d for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23]=-=[24]-=-[26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. With emp... |

8 |
Characterizing topologies with bounded complete computational models
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(Show Context)
Citation Context ...as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11]=-=[5]-=-[12][23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from... |

8 | Domain theoretic models of topological spaces
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(Show Context)
Citation Context ...cated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12]=-=[23]-=-[24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. With... |

7 |
Computation on metric spaces via domain theory. Topology and its applications, 85:247–263
- Lawson
- 1998
(Show Context)
Citation Context ...s dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17]=-=[18]-=-[11][5][12][23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherit... |

6 | Uniform approximation of topological spaces
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(Show Context)
Citation Context ...indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5]=-=[12]-=-[23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. ... |

6 |
Total objects of domains
- Kamimura, Tang
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(Show Context)
Citation Context ...ld be executed in a purely domain theoretic manner ([6][7][8][9]), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang =-=[15]-=-, and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23][24][26][27][19][28][3]). A model of a space X is a ... |

6 | The space of maximal elements of a compact domain, Electonic
- Martin
(Show Context)
Citation Context ...r instance by Scott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23][24]=-=[26]-=-[27][19][28][3]). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. With emphasi... |

5 | General topology. Polish Scienti - Engelking - 1977 |

4 | A Category of Compositional Domain-Models for Separable Stone Spaces
- Alessi, Baldan, et al.
- 2003
(Show Context)
Citation Context ...ott [29], Kamimura and Tang [15], and Abramsky [1]. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23][24][26][27][19][28]=-=[3]-=-). A model of a space X is a continuous dcpo D for which there is a homeomorphism between X and the space of maximal elements max(D) in its Scott topology inherited from D. With emphasis: (a) The topo... |

3 | A principle of induction - Martin - 2001 |

2 |
Total objects in domains. Unpublished notes
- Abramsky
- 1985
(Show Context)
Citation Context ... purely domain theoretic manner ([6][7][8][9]), there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29], Kamimura and Tang [15], and Abramsky =-=[1]-=-. But since this realization, one gets the distinct impression that interest in this topic may be growing ([17][18][11][5][12][23][24][26][27][19][28][3]). A model of a space X is a continuous dcpo D ... |

2 | A renee equation for algorithmic complexity
- Martin
- 2001
(Show Context)
Citation Context ...n induction principle discussed in [21] works especially nice on ideal domains. A splitting on a domain is a selfmap above the identity. They are used to model the recursive steps in algorithms ([20],=-=[22-=-],[25]). To do so, we must know which splittings on a domain have thesxed point property: That is, if s : D ! D is a splitting, we want to know when it is that G n0 s n (x) 2sx(s); for all x 2 D; wher... |