## Continuous Lattices and DomainsInformation content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. The authors develop the mathematical foundations of partially ordered sets with completeness properties of various degrees, in particular directed complete ordered sets and complete lattices. Uniquely, they focus on partially ordered sets that have an extra order relation, modelling the notion that one element 'finitely approximates' another, something closely related to intrinsic topologies linking order and topology. Extensive use is made of topological ideas, both by defining useful topologies on the structures themselves and by developing close connections with numerous aspects of topology. The theory so developed not only has applications to computer science but also within mathematics to such areas as analysis, the spectral theory of algebras and the theory of computability. This authoritative, comprehensive account of the subject will be essential for all those working in the area. |

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### Contents

A Primer on Ordered Sets and Lattices | 1 |

Exercises | 7 |

The Scott Topology | 131 |

The Lawson Topology | 208 |

Morphisms and Functors | 264 |

Functors | 318 |

Spectral Theory of Continuous Lattices | 394 |

Compact Posets and Semilattices | 439 |

Applications | 492 |

Exercises | 521 |

Articles | 528 |

559 | |

List of Symbols | 568 |

575 | |

### Common terms and phrases

algebraic apply approximating arbitrary assume basis bounded complete called Chapter characterization closure complete lattice complete semilattice Computer conclude consider consisting construction containing continuous lattice convergence Corollary dcpo defined Definition denote directed set directed sups distributive domain dual element equal equation equivalent example Exercise exists fact finite function functor given Hausdorff hence Hofmann ideal implies infs injective intersection interval irreducible isomorphism Lawson topology least Lemma limit locally compact lower adjoint Mathematics maximal meet continuous monotone morphisms natural neighborhood nonempty Note objects open filter operation particular poset preceding preserves prime projective Proof Proposition Prove relation Remark respect satisfies saturated Scott topology Scott-continuous semilattice sober space Spec specialization strict subset sup semilattice Suppose supremum Theorem theory topological semilattice topological space union unique upper set