Hofmann-Mislove type definitions of non-Hausdorff spaces

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2022-01-01 DOI:10.1017/S0960129522000196
Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao
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Abstract

Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered.
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非Hausdorff空间的Hofman-Mislove型定义
摘要域理论中最重要的结果之一是Hofmann Mislove定理,它通过开滤子揭示了清醒空间的一个非常独特的特征。在本文中,我们将这个结果推广到d-空间和良好过滤空间。我们通过引入Hofmann-Mislove系统(简称HM系统)和$\Psi$-良好过滤空间的概念来实现这一点,它们为清醒空间、良好过滤空间和d空间提供了一种新的统一方法。此外,通过$\Psi$集提供了$\Psi$-良好过滤空间的特征化。我们还讨论了Xu所考虑的$\Psi$-良好滤波空间与H-ober空间之间的关系。我们证明了完备$\Psi$-良好过滤空间的范畴是具有连续映射的$T_0$空间范畴的全反射子范畴。对于每个具有指定属性的HM系统$\Psi$,我们证明了$T_0$空间X是$\Psi$-良好过滤的当且仅当其Smyth幂空间$P_s(X)$是$\Psi$-良好滤波的。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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