The first-countability in generalizations of topological groups with ideal convergence

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-02-01 DOI:10.1016/j.topol.2024.109150

Xin Liu, Shou Lin, Xiangeng Zhou

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引用次数: 0

Abstract

The study of convergence in topological groups has become a frontier research subject. In many cases, the first-countability is an important and strong condition. Based on

I_{s n}

-continuity, the present paper discusses how topology and algebra are related through a notion of continuity generated by ideal convergence. We introduce the classes of generalizations of topological groups, give the structures of

I_{s n}

-topological groups by certain sequential coreflections, and obtain generalized metric properties of

I

-snf-countable para-

I_{s n}

-topological groups.

Let

I

be an admissible ideal on the set

N

of natural numbers. The following results are obtained.

(1)
Every T₂, $I$ -snf-countable para- $I_{s n}$ -topological group is an sn-quasi-metrizable and cs-submetrizable space.
(2)
A T₀, $I_{s n}$ -topological group is an $I$ -snf-countable space if and only if it is a cs-metrizable space satisfying that each sequentially open subset is $I_{s n}$ -open.

These show the unique role of

I_{s n}

-continuity in the study of topological groups and related structures, and present a version of topological algebra using the notion of ideals.

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具有理想收敛的拓扑群的推广中的第一可数性

拓扑群的收敛性研究已成为一个前沿研究课题。在许多情况下，第一可数性是一个重要且强有力的条件。本文在isn -连续性的基础上，通过理想收敛所产生的连续性概念，讨论了拓扑与代数之间的关系。引入了拓扑群的推广类，给出了isn -拓扑群的序共反射结构，得到了i -snf可数的拟isn -拓扑群的广义度量性质。设我是自然数集合N上的一个可容许理想。得到了以下结果：(1)每个T2， i- snf-可数的拟isn -可数拓扑群是一个sn-拟可度量和cs-子可度量的空间。(2)T0， isn -拓扑群是一个cs-可度量空间，当且仅当它是一个cs-可度量空间，满足每个序开子集是isn -开的。这表明了isn -连续性在拓扑群和相关结构研究中的独特作用，并提出了一种使用理想概念的拓扑代数。

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.