D-completions and the d-topology
D-completions and the d-topology
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DOI：
10.1016/j.apal.2008.06.019
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发表时间：
2009-06
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在这篇文章中，我们通过反射函子给出了T0-空间的各种完备化的一般范畴构造，其中特别强调了我们所称的D-完备化，这是由Wyler[O.Wyler，DedekindComplete Poset and Scott Topology，in：B.Banaschewski，R.-E.Hoffmann(编辑)，Continuous Lateses Proceaughes，Bremen1979，in：Lesson Note in MATHEMICS，Vol.871，Springer Verlag，1981，pp.384-389]提出的一种有向完备化的类型。一个关键的结果是，特定类型的所有补全都是通用的，因此是唯一的(直到同胚)。我们给出了D-完备的直接定义，并通过引入Scott拓扑的一个变体来发展它的理论，我们称之为d-拓扑。对于偏序集，D-完成被证明是一个自然的dcpo-完成，它推广了四舍五入的理想完成。在本文的后半部分，我们分别考虑了D-完备化和闭合理想完备化的设置。
In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the D-completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384–389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D-completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the D-completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the D-completion agrees with the sobrification respectively the closed ideal completion.