D-completion, well-filterification and sobrification

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-08-26 DOI:10.1016/j.topol.2024.109050

引用次数: 0

Abstract

In this paper, we study the D-completion, well-filterification and sobrification of a $T_{0}$ space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set. In 2020, Xu et al. asked whether closed RD-sets are exactly closed WD-sets for every $T_{0}$ space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed WD-set. Second, we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements. Finally, for a given $T_{0}$ space, we give some sufficient conditions which guarantee that each pair of its standard D-completion, standard well-filterification and standard sobrification agrees.

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D 级完井、滤井和净化

在本文中，我们研究了 T0 空间的 D-补全、井滤化和索布尔化。首先，我们举例说明既不是有向集的闭集也不是闭 KF 集的锥形闭集。2020 年，Xu 等人提出了这样一个问题：对于每个 T0 空间，闭 RD 集是否都是完全闭的 WD 集？这个例子也给出了上述问题的否定答案，因为每个锥形闭集都是一个闭 WD 集。其次，我们利用前 C-紧凑元素的概念，为正集的 D-补集提供了一个直接表征。最后，对于给定的 T0 空间，我们给出了一些充分条件，以保证其标准 D-补集、标准井滤波和标准净化的每一对都是一致的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.