A few projective classes of (non-Hausdorff) topological spaces

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-07-09 DOI:10.1016/j.topol.2024.109009

Jean Goubault-Larrecq

引用次数: 0

Abstract

A class of topological spaces is projective (resp., ω-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, (ω-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even ω-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.

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非豪斯多夫）拓扑空间的几个投影类

一类拓扑空间是投影的（或者说，-投影的），当且仅当类中空间的投影系统（或者说，具有可数同尾子集的指数）仍然在类中。已知一定数量的豪斯多夫空间类是或不是（-）射影的。我们将研究不一定是 Hausdorff 空间的类。清醒空间和紧凑清醒空间构成了投影类，但大多数局部紧凑空间类甚至不是-投影的。稳定紧凑空间正是局部紧凑的强清醒空间，而强清醒空间正是清醒、相干、紧凑、弱 Hausdorff（在 Keimel 和 Lawson 的意义上）空间，在这一事实的指导下，我们研究了由这些性质的组合定义的哪些类是射影的。值得注意的是，我们发现相干清醒空间、紧凑相干清醒空间以及（局部）强清醒空间构成了射影类。

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.