产品的相对功能可数子集

IF 0.6 4区数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-11-12 DOI:10.1016/j.topol.2024.109133

Anton E. Lipin

引用次数: 0

摘要

如果对于每个连续函数 f:X→R 的集合 f[A] 是可数的，那么拓扑空间 X 的子集 A 称为 X 中的相对函数可数（RFC）。我们假定空间 Xn 是 Tychonoff 的，且每个 Xn 的所有 RFC 子集都是可数的，从而证明乘积 ∏n∈ωXn 的所有 RFC 子集都是可数的。证明的主要工具是下面的结果：对于每一个Tychonoff空间X和任何可数集Q⊆X，有一个连续函数f:Xω→R2，使得f对Qω的限制是注入的。

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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Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function

f : X \to R

the set

f [A]

is countable. We prove that all RFC subsets of a product

\prod_{n \in ω} X_{n}

are countable, assuming that spaces

X_{n}

are Tychonoff and all RFC subsets of every

X_{n}

are countable. In particular, in a metrizable space every RFC subset is countable.

The main tool in the proof is the following result: for every Tychonoff space X and any countable set

Q \subseteq X

there is a continuous function

f : X^{ω} \to R^{2}

such that the restriction of f to

Q^{ω}

is injective.

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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.

期刊最新文献

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