## The Regular Spaces With Countably Based Models

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 4 - 2 self |

### BibTeX

@ARTICLE{Martin_theregular,

author = {Keye Martin},

title = {The Regular Spaces With Countably Based Models},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {}

}

### OpenURL

### Abstract

The regular spaces which may be realized as the set of maximal elements in an !-continuous dcpo are the Polish spaces. In addition, we give a new and conceptually simple model for complete metric spaces. These results enable

### Citations

479 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ... decided that any separable metric space with a model had to be Polish. This paper is about the proof of that aesthetical imperative. 2 Background 2.1 Domain Theory A poset is a partially ordered set =-=-=-[1]. 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... |

376 | Classical Descriptive Set Theory - Kechris - 1995 |

45 | A computational model for metric spaces
- Edalat, Heckmann
- 1998
(Show Context)
Citation Context ...Implicit in this result is thesrst proof that a Polish space could be modelled, but the complexity of his construction does not lend much insight into why. This was provided by Edalat and Heckmann in =-=[-=-5]. Example 2.13 A model for complete metric spaces. Given a metric space (X; d), the formal ball model [5] BX = X [0; 1) is a poset when ordered via (x; r) v (y; s) , d(x; y) r s: The approximation... |

40 | A Foundation for Computation
- Martin
- 2000
(Show Context)
Citation Context ... f(S): We will need the next result in our construction of a model for complete metric spaces. The set [0; 1) denotes the domain of nonnegative reals in their opposite order. 3 Theorem 2.10 (Martin [11]) Let : P ! [0; 1) be a map on a poset P which is strictly monotone: x v y & x 6= y ) x > y. If every increasing sequence in P has a supremum preserved by , then (i) P is a dcpo, (ii) is Scot... |

36 |
Spaces of maximal points
- Lawson
- 1997
(Show Context)
Citation Context ...meomorphism : X ! maxD where maxD carries its relative Scott topology inherited from D. If in addition the domain D is !-continuous, then (D; : X ' maxD) is called a countably based model. Lawson [1=-=0] prov-=-ed that a certain subset of countably based models captured exactly the Polish spaces. For a domain D, we say that the relative Scott and Lawson topologies on maxD agree if "x \ maxD is a Scott c... |

28 | The measurement process in domain theory
- Martin
- 2000
(Show Context)
Citation Context ...6.3 A measurement on a domain D is a Scott continuous map : D ! [0; 1) with ! ker where ker = fx 2 D : x = 0g: A simple introduction to measurement and its basic applications is given in [13], where a proof of the following can be found. Lemma 6.4 (Martin [11]) If is a measurement on a domain D, then ker maxD: We dene the following classes of countably based domains: T 3 is the c... |

16 | When Scott is weak at the top
- Edalat
- 1997
(Show Context)
Citation Context ...h such a model must be Polish by Corollary 4.7. For the converse, we simply use the formal ball model. 2 5 The Probabilistic Powerdomain All denitions in this section are identical to those given in [=-=4-=-]. We try to be brief for the sake of space. Denition 5.1 The probabilistic powerdomain of a domain D is the set PD of all continuous valuations in the pointwise order. The normalised probabilistic po... |

11 |
A Glimm{Eros dichotomy for Borel equivalence relations
- Harrington, Kechris, et al.
- 1990
(Show Context)
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 [7], while the others are all due to Choquet [2]. Theorem 4.3 The Scott topology on a domain is Choquet complete. Proof We dene the approximation scheme a : f(U; x) : x 2 U 2 D g ! D as follows: Given... |

8 |
Characterizing topologies with bounded complete computational models
- Ciesielski, Flagg, et al.
- 1999
(Show Context)
Citation Context ...ls are the Polish spaces. Thus, the best we can hope for is a model of Polish spaces and continuous mappings. Luckily, this much is possible, and in fact, we can use bounded complete domains to do so =-=[3]-=-, which means that continuous maps on Polish spaces will have greatest Scott continuous extensions. In addition, the countably based model problem for zero-dimensional spaces is now solved. Corollary ... |

6 | The space of maximal elements of a compact domain, Electonic - Martin |

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

5 | Ideal models of spaces
- Martin
(Show Context)
Citation Context ...axD is a Gssubset of D. However, maxD is not even a Hausdor space: The sequence (m n ) converges to both 1 1 and 1 2 . The spaces modelled by objects in Tscan be understood in terms of ideal domains [=-=12-=-]. Whether T = Tsor Ts= T 1 is unknown and the author expects nontrivial. 7 Questions (i) Can the rationals Q be embedded as a closed subset of maxD for D !-continuous? (ii) Is maxD a Gssubset of D w... |

2 | Domain Environments - Heckmann |