Quasicontinuous Domains and the Smyth Powerdomain
Quasicontinuous Domains and the Smyth Powerdomain
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DOI：
10.1016/j.entcs.2013.09.015
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发表时间：
2013-11
期刊：
影响因子：
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通讯作者：
R. Heckmann;K. Keimel
中科院分区：
文献类型：
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作者：
R. Heckmann;K. Keimel
摘要在Domain理论中，准连续Domain的出现使连续Domain这个强有力的概念有了一点推广。本文的目的是表明，拟连续域发生在一个自然的方式有关的powersomains生成和紧饱和子集。从这个观点出发，似乎可以最好地理解拟连续域的性质。这是在对比以前的方法，其中一个quasicontinuous域的性质进行了比较，主要是与斯科特开子集的格的属性。我们提出了一个特征的域，出现的域的非空紧饱和子集的拟连续域。由于ME Rudin的一个集合论引理在拟连续域的发展中起到了至关重要的作用。我们提出了一个拓扑变形的鲁丁的引理不可约集取代定向集。不可约集的概念在这里是指一个非空集不能被两个闭集覆盖，除非其中一个集合已经覆盖了它。由于有向集是偏序集上Alexandroff拓扑的不可约集，这是一个自然的推广。它允许一个显着的特点清醒的空间。为此，我们用Q X表示拓扑空间X的非空紧饱和子集空间（具有上Vietoris拓扑）。下列性质是等价的：（1）X是sober，（2）Q X是sober，（3）X在以下意义下是强良滤子：只要A是Q X的不可约子集，U是X的开子集，使得<$A <$U，则对某个K∈ A，K <$U。这一结果填补了现有文献中的空白。
Abstract In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. This is in contrast to the previous approaches where the properties of a quasicontinuous domain were compared primarily with the properties of the lattice of Scott-open subsets. We present a characterization of those domains that occur as domains of nonempty compact saturated subsets of a quasicontinuous domain. A set theoretical lemma due to ME Rudin has played a crucial role in the development of quasicontinuous domains. We present a topological variant of Rudinʼs Lemma where irreducible sets replace directed sets. The notion of irreducibility here is that of a nonempty set that cannot be covered by two closed sets except if already one of the sets is covering it. Since directed sets are the irreducible sets for the Alexandroff topology on a partially ordered set, this is a natural generalization. It allows a remarkable characterization of sober spaces. For this we denote by Q X the space of nonempty compact saturated subsets (with the upper Vietoris topology) of a topological space X. The following properties are equivalent:(1) X is sober,(2) Q X is sober,(3) X is strongly well-filtered in the following sense: Whenever A is an irreducible subset of Q X and U an open subset of X such that⋂ A⊆ U, then K⊆ U for some K∈ A. This result fills a gap in the existing literature.