Velichko's notions close to sequential separability and their hereditary variants in Cp-theory

IF 0.6 4区数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-10-02 DOI:10.1016/j.topol.2024.109076

Alexander V. Osipov

引用次数: 0

Abstract

A space X is sequentially separable if there is a countable

S \subset X

such that every point of X is the limit of a sequence of points from S. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: σ-separability and F-separability. The aim of this paper is to study σ-separability and F-separability (and their hereditary variants) of the space

C_{p} (X)

of all real-valued continuous functions, defined on a Tychonoff space X, endowed with the pointwise convergence topology. In particular, we proved that σ-separability coincides with sequential separability. Hereditary variants (hereditarily σ-separability and hereditarily F-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.

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维利奇科的接近于顺序可分性的概念及其在 Cp 理论中的遗传变异

如果存在一个可数 S⊂X，使得 X 的每个点都是来自 S 的点序列的极限，则空间 X 是顺序可分的。2004 年，N.V. Velichko 定义并研究了与顺序可分性接近的概念：σ 可分性和 F 可分性。本文的目的是研究所有实值连续函数空间 Cp(X) 的 σ 可分性和 F 可分性（及其遗传变异），这些函数定义在泰克诺夫空间 X 上，并赋有点收敛拓扑。我们特别证明了 σ 可分性与顺序可分性重合。在宇宙空间类中，遗传变异（遗传σ可分性和遗传F可分性）与弗雷谢特-乌里索恩性质重合。

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.

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