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A study on some versions of mi-spaces

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-06-30 DOI:10.1016/j.topol.2025.109493

Liang-Xue Peng

引用次数: 0

Abstract

We introduce notions of c-σ-

m_{i}

-spaces (c-

m_{i}

-spaces) for

i \in {1, 2, 3}

. A space X is called a c-

m_{1}

-space (c-

m_{2}

-space, c-

m_{3}

-space) if X has a closure-preserving local base (closure-preserving local quasi-base, cushioned local pair-base) at every compact subset F of X. We give some characterizations of σ-

m_{i}

-spaces (

m_{i}

-spaces), s-σ-

m_{i}

-spaces (s-

m_{i}

-spaces), and c-σ-

m_{i}

-spaces (c-

m_{i}

-spaces). Thus some known conclusions can be obtained by these characterizations. We also get the following results. Every stratifiable

T_{1}

-space is a c-

m_{2}

-space. Every ordinal is hereditarily a c-

m_{1}

-space. If X is a generalized ordered (GO-) space and a sequence

{x_{n}}_{n \in N}

of points in X has a limit point x in X, then the set

C = {x_{n} : n \in N} \cup {x}

has a closure-preserving local base in X. If X is a monotonically (countably) metacompact regular space, then X is a

m_{3}

-space. If

X_{n}

is a regular space which has a σ-NSR pair-base at every point of

X_{n}

for each

n \in N

, then the product space

X = \prod_{n \in N} X_{n}

is a σ-

m_{3}

-space. If

X_{n}

is a space which has a σ-NSR pair-base at a compact subset

C_{n}

X_{n}

for each

n \in N

, then the product space

X = \prod_{n \in N} X_{n}

has a σ-NSR pair-base at the set

C = \prod_{n \in N} C_{n}

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若干版本的半空间研究

对于i∈{1,2,3}，我们引入c-σ-mi-spaces （c-mi-spaces）的概念。如果X在X的每一个紧子集F上都有一个保持闭的局部基（保持闭的局部拟基，缓冲的局部对基），则空间X称为c-m1-空间（c-m2-空间，c-m3-空间）。我们给出了σ-mi-空间（mi-spaces）、s-σ-mi-空间（s-mi-spaces）和c-σ-mi-空间（c-mi-spaces）的一些刻画。因此，通过这些表征可以得出一些已知的结论。我们还得到以下结果。每个可分层的t1空间都是c-m2空间。每个序数都是一个c-m1空间。若X是广义有序（GO-）空间，且X中点的序列{xn}n∈n在X中有极限点X，则集合C={xn:n∈n}∪{X}在X中有闭保局部基。若X是单调（可数）元紧正则空间，则X是一个m3空间。若Xn是一个正则空间，对于n∈n，在Xn的每一点上都有一个σ-NSR对基，则积空间X=∏n∈NXn是一个σ-m3空间。如果Xn是一个空间，对于每个n∈n在Xn的紧子集Cn上有一个σ-NSR对基，则乘积空间X=∏n∈NXn在集合C=∏n∈NCn上有一个σ-NSR对基。

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.