关于几乎离散的格罗内迪克空间的乘积

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-04-17 DOI:10.1016/j.topol.2024.108919

Alexander V. Osipov

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引用次数: 0

摘要

如果拓扑空间 X 恰好有一个非孤立点，则称其为近乎离散空间。在本文中，我们得到，对于几乎离散空间 Xi 的可数乘积 X=∏Xi 所有连续实值函数的空间 Cp(X) 是一个 μ 空间，当且仅当且仅当 t(X)=ω 时，X 是一个弱 q 空间，当且仅当且仅当 X 是由其所有可数子空间的族函数生成的。这一结果使得解决关于格罗内狄克空间乘积的阿昌吉利问题成为可能。证明了在通过添加一个科恩实数得到的 ZFC 模型中，存在格罗内狄克空间 X 和 Y，使得 X×Y 不是弱格罗内狄克空间。在（PFA）中：任何可数族几乎离散的格罗内狄克空间的乘积都是格罗内狄克空间。

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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On the product of almost discrete Grothendieck spaces

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product $X = \prod X_{i}$ of almost discrete spaces $X_{i}$ the space $C_{p} (X)$ of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, $t (X) = ω$ if, and only if, X is functionally generated by the family of all its countable subspaces.

This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that $X \times Y$ is not weakly Grothendieck space. In $(P F A)$ : the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

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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.