{"title":"On the product of almost discrete Grothendieck spaces","authors":"Alexander V. Osipov","doi":"10.1016/j.topol.2024.108919","DOIUrl":null,"url":null,"abstract":"<div><p>A topological space <em>X</em> is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product <span><math><mi>X</mi><mo>=</mo><mo>∏</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> of almost discrete spaces <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> the space <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of all continuous real-valued functions with the topology of pointwise convergence is a <em>μ</em>-space if, and only if, <em>X</em> is a weak <em>q</em>-space if, and only if, <span><math><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>ω</mi></math></span> if, and only if, <em>X</em> is functionally generated by the family of all its countable subspaces.</p><p>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 <em>ZFC</em>, obtained by adding one Cohen real, there are Grothendieck spaces <em>X</em> and <em>Y</em> such that <span><math><mi>X</mi><mo>×</mo><mi>Y</mi></math></span> is not weakly Grothendieck space. In <span><math><mo>(</mo><mi>P</mi><mi>F</mi><mi>A</mi><mo>)</mo></math></span>: the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864124001044","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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 of almost discrete spaces the space 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, 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 is not weakly Grothendieck space. In : the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.
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.