The reflectivity and reflective hull of closure space categories

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-03-18 DOI:10.1016/j.topol.2025.109345

Zhongxi Zhang

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引用次数: 0

Abstract

The notion of a reflective subcategory provides a convenient way in dealing with various types of completions. This paper investigates the reflectivity of full subcategories in the category CL₀ of

T_{0}

closure spaces, specifically those containing pointed non-

T_{1}

-spaces. One key result is that such a category is reflective in CL₀ if and only if it is the category of all Z-convergence spaces for some subset system Z. Leveraging this result, we offer a unified form for their reflective hulls in CL₀. Using similar techniques, we establish a unified form for the reflective hull of full subcategories in the category TOP₀ of

T_{0}

topological spaces, including at least one non-

T_{1}

-space. In light of this, we demonstrate that the reflective hull of the category KBSOB of k-bounded sober spaces within TOP₀ is TOP₀ itself.

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封闭空间的反射率和反射壳类

反射子类别的概念为处理各种类型的补全提供了一种方便的方法。本文研究了T0闭包空间，特别是包含点非t1空间的闭包空间CL0的满子范畴的反射率。一个关键的结果是，当且仅当它是某些子集系统z的所有z收敛空间的范畴时，这样的范畴在CL0中是反射的。利用这个结果，我们提供了CL0中它们的反射壳的统一形式。利用类似的技术，我们建立了T0拓扑空间的TOP0范畴中包含至少一个非t1空间的全子范畴反射壳的统一形式。据此，我们证明了TOP0内k有界清醒空间的类别KBSOB的反射壳就是TOP0本身。

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.