## A Hofmann-Mislove Theorem for Bitopological Spaces (2008)

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### BibTeX

@MISC{Jung08ahofmann-mislove,

author = {Achim Jung and et al.},

title = {A Hofmann-Mislove Theorem for Bitopological Spaces},

year = {2008}

}

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### Citations

1065 | Introduction to Lattices and Order - Davey, Priestley - 2002 |

456 | Domain Theory - Abramsky, Jung - 1994 |

381 | Bilattices and the semantics of logic programming - Fitting - 1991 |

292 |
A useful four-valued logic
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- 1977
(Show Context)
Citation Context ...����� as the logical structure of a d-frame. Altogether, then, we see that d-frames are special “bilattices,” which were introduced by Ginsberg, [9], as a generalisation of Belnap’s four-valued logic =-=[4]-=-. Exploiting Proposition 3.1 we can easily compute conjunction and disjunction in terms of the representation of a d-frame as � ��� � � : � ����� � ��� � � ��� ������� � ������� � � ������� � � ������... |

228 | Domain theory in logical form - Abramsky - 1991 |

69 |
Powerdomains and predicate transformers: a topological view
- Smyth
- 1983
(Show Context)
Citation Context ...rame of opens) to the idea of the co-compact topology on the space, [16]. Its significance in Computer Science took some time to emerge, and credit in this respect is due to Plotkin, [17, 18], Smyth, =-=[19]-=-, and Vickers, [21], who pointed out that it is at the core of the proof that the upper powerdomain (defined as a free algebraic theory) has a concrete representation as a set of subsets of the given ... |

60 |
Post-graduate lecture notes in advanced domain theory
- Plotkin
- 1981
(Show Context)
Citation Context ...(applied to the frame of opens) to the idea of the co-compact topology on the space, [16]. Its significance in Computer Science took some time to emerge, and credit in this respect is due to Plotkin, =-=[17, 18]-=-, Smyth, [19], and Vickers, [21], who pointed out that it is at the core of the proof that the upper powerdomain (defined as a free algebraic theory) has a concrete representation as a set of subsets ... |

56 | Multivalued logics: A uniform approach to inference in artificial intelligence
- Ginsberg
- 1992
(Show Context)
Citation Context ...frame, and suggests that we regard ��������� � ������������� as the logical structure of a d-frame. Altogether, then, we see that d-frames are special “bilattices,” which were introduced by Ginsberg, =-=[9]-=-, as a generalisation of Belnap’s four-valued logic [4]. Exploiting Proposition 3.1 we can easily compute conjunction and disjunction in terms of the representation of a d-frame as � ��� � � : � �����... |

46 |
Local compactness and continuous lattices
- Hofmann, Mislove
(Show Context)
Citation Context ...s and show that these are always spatial. Keywords: Bitopological spaces, d-frames, Stone duality, sober spaces, Hofmann-Mislove theorem 1 Introduction The Hofmann-Mislove theorem, first published as =-=[10]-=-, states that in a sober space the open neighbourhood filters of compact saturated sets are precisely the Scott-open filters in the corresponding frame of opens. Mathematically, it has some remarkable... |

35 | Spaces, volume 3 of Cambridge - Stone - 1982 |

34 | Topology via logic, volume 5 of Cambridge Tracts - Vickers - 1989 |

28 |
Domain Theory In Logical Form. Annals of Pure and Applied logic
- Abramsky
- 1988
(Show Context)
Citation Context ...ne naturally has the upper topology generated by sets of the form �������¨������¦�����§���� an open in the original space), and the lower topology generated by sets of the form � (� ����� . Abramsky, =-=[1]-=-, showed that the three powerdomains ������������¦���������� can be obtained systematically from this (bi-)topological point of view. Our interest in bitopological spaces was driven by these examples ... |

28 | Cartesian Closed Categories of Domains, volume 66 of CWI Tracts. Centrum voor Wiskunde en Informatica - Jung - 1989 |

26 | Synthetic topology of data types and classical spaces
- Escardó
(Show Context)
Citation Context ...puter Science, The University of Birmingham, Birmingham, United ¥ Kingdom Department of Mathematics & Computer Science, Chapman University, Orange, CA, USA 1sMore recently, Taylor, [20], and Escardó, =-=[6]-=-, have interpreted the theorem as expressing the idea that the compact saturated sets are precisely those for which there is a continuous universal quantifier. To this end, they read “open set” as “pr... |

24 |
Stone Spaces. Cambridge
- Johnstone
- 1982
(Show Context)
Citation Context ...d we refer the reader to the report [13]. 2 Stone duality and the Hofmann-Mislove theorem We briefly review the duality between topological spaces and frames. For more details see [2, Chapter 7], and =-=[11, 8]-=-. Definition 2.1 A frame is a complete lattice in which finite meets distribute over arbitrary joins. We denote with � , � , � , � , and � the order, finite meets, arbitrary joins, least and largest e... |

22 | Totale Objekte und Mengen in der Bereichstheorie - Berger - 1990 |

19 |
Dijkstra’s predicate transformers and Smyth’s powerdomains
- Plotkin
- 1980
(Show Context)
Citation Context ...(applied to the frame of opens) to the idea of the co-compact topology on the space, [16]. Its significance in Computer Science took some time to emerge, and credit in this respect is due to Plotkin, =-=[17, 18]-=-, Smyth, [19], and Vickers, [21], who pointed out that it is at the core of the proof that the upper powerdomain (defined as a free algebraic theory) has a concrete representation as a set of subsets ... |

18 |
A direct proof of the Hofmann-Mislove theorem
- Keimel, Paseka
- 1994
(Show Context)
Citation Context ...to state. Theorem 2.4 In a sober space ��������� , there is a bijection between the set of compact saturated subsets of � and the set of Scott-open filters in � . Although a direct proof is possible, =-=[15]-=-, it more useful for us to refer to Stone duality, as in the original paper [10]: Lemma 2.5 A Scott-open filter in a frame � is equal to the intersection of the collection of completely prime filters ... |

15 |
A ternary operation in distributive lattices
- Birkhoff, Kiss
- 1947
(Show Context)
Citation Context ...dditional relation is always present in a d-frame, and in fact it follows from the distributive lattice structure and the two complemented elements alone. The earliest reference to this appears to be =-=[5]-=-, but the proof is entirely straightforward and can be left as an exercise. Proposition 4.1 Let ��������������������� be a bounded distributive lattice, and ��������� a complemented pair in � , that i... |

13 | Infinite sets that admit fast exhaustive search - Escardó - 2007 |

12 | On the duality of compact vs. open
- Jung, Sünderhauf
- 1996
(Show Context)
Citation Context ... Theorem 6.5 above, and this part of the story works out in a most satisfying way. Another motivation was the desire to extend the duality between stably compact spaces and strong proximity lattices, =-=[14]-=-. There, it is the case that the two topologies 15sdetermine each other (each being the co-compact topology with respect to the other), but this is no longer true in the locally compact case: Example ... |

12 |
The duality of continuous posets
- Lawson
- 1979
(Show Context)
Citation Context ...pact saturated subsets of a sober space form a dcpo when ordered by reverse inclusion, and it links Lawson duality (applied to the frame of opens) to the idea of the co-compact topology on the space, =-=[16]-=-. Its significance in Computer Science took some time to emerge, and credit in this respect is due to Plotkin, [17, 18], Smyth, [19], and Vickers, [21], who pointed out that it is at the core of the p... |

11 |
Topology Via Logic,” Cambridge Tracts
- Vickers
- 1989
(Show Context)
Citation Context ...he idea of the co-compact topology on the space, [16]. Its significance in Computer Science took some time to emerge, and credit in this respect is due to Plotkin, [17, 18], Smyth, [19], and Vickers, =-=[21]-=-, who pointed out that it is at the core of the proof that the upper powerdomain (defined as a free algebraic theory) has a concrete representation as a set of subsets of the given domain. Quite unexp... |

10 |
Cartesian Closed Categories of Domains,” CWI Tracts 66, Centrum voor Wiskunde en Informatica
- Jung
- 1989
(Show Context)
Citation Context ...free algebraic theory) has a concrete representation as a set of subsets of the given domain. Quite unexpectedly, it was also required in the classification of cartesian closed categories of domains, =-=[12]-=-. Published in the Proceedings of the 23rd Annual Conference on Mathematical Foundations of Program£ ming Semantics (MFPS XXIII), eds M. Fiore and M. Mislove, Electronic Notes in Theoretical Computer ... |

10 | Operational domain theory and topology of a sequential programming language - Escardó, Ho - 2005 |

6 | Geometric and higher order logic in terms of abstract Stone duality, Theory and Applications of Categories 7
- Taylor
- 2000
(Show Context)
Citation Context ...?-??. School of Computer Science, The University of Birmingham, Birmingham, United ¥ Kingdom Department of Mathematics & Computer Science, Chapman University, Orange, CA, USA 1sMore recently, Taylor, =-=[20]-=-, and Escardó, [6], have interpreted the theorem as expressing the idea that the compact saturated sets are precisely those for which there is a continuous universal quantifier. To this end, they read... |

5 |
On the bitopological nature of Stone duality
- Jung, Moshier
- 2006
(Show Context)
Citation Context ...gical spaces was driven by these examples and also by a desire to analyse various Stone dualities, but there is no room here to expand on this latter aspect; instead we refer the reader to the report =-=[13]-=-. 2 Stone duality and the Hofmann-Mislove theorem We briefly review the duality between topological spaces and frames. For more details see [2, Chapter 7], and [11, 8]. Definition 2.1 A frame is a com... |