Preservation and reflection of separation axioms by essentially Kolmogorov and Kolmogorov relations

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-09-03 DOI:10.1016/j.topol.2025.109574

Jeffrey T. Denniston , Stephen E. Rodabaugh , Jamal K. Tartir

{"title":"Preservation and reflection of separation axioms by essentially Kolmogorov and Kolmogorov relations","authors":"Jeffrey T. Denniston , Stephen E. Rodabaugh , Jamal K. Tartir","doi":"10.1016/j.topol.2025.109574","DOIUrl":null,"url":null,"abstract":"<div><div>This paper focuses on the Kolmogorov functor <span><math><mi>K</mi><mo>:</mo><mrow><mi>Top</mi></mrow><mo>→</mo><msub><mrow><mi>Top</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and associated ideas. There are two main objectives: first, catalogue and prove those topological invariants which <em>K</em> both preserves and reflects, called “Hong” invariants; and second, give a step-by-step axiomatic foundation for <em>K</em> to analyze its remarkable success in having so many Hong invariants. Pursuing the second objective leads to “essentially Kolmogorov” (EK) relations, the family of which on a ground set forms a complete lattice ordered by inclusion; the diagonal relation Δ is the universal lower bound and the Kolmogorov relation <em>K</em> is the universal upper bound—typically there are many EK relations strictly between Δ and <em>K</em>. Though EK relations are significant weakenings of <em>K</em>, they enjoy the same success w.r.t. Hong invariants. Counterexamples clarify relationships between similar notions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109574"},"PeriodicalIF":0.5000,"publicationDate":"2025-09-03","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/S0166864125003724","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

This paper focuses on the Kolmogorov functor

K : Top \to {Top}_{0}

and associated ideas. There are two main objectives: first, catalogue and prove those topological invariants which K both preserves and reflects, called “Hong” invariants; and second, give a step-by-step axiomatic foundation for K to analyze its remarkable success in having so many Hong invariants. Pursuing the second objective leads to “essentially Kolmogorov” (EK) relations, the family of which on a ground set forms a complete lattice ordered by inclusion; the diagonal relation Δ is the universal lower bound and the Kolmogorov relation K is the universal upper bound—typically there are many EK relations strictly between Δ and K. Though EK relations are significant weakenings of K, they enjoy the same success w.r.t. Hong invariants. Counterexamples clarify relationships between similar notions.

查看原文本刊更多论文

柯尔莫哥洛夫和柯尔莫哥洛夫关系对分离公理的保存和反映

本文主要研究Kolmogorov函子K:Top→Top0及其相关思想。主要有两个目标：第一，列出并证明K既保留又反映的拓扑不变量，称为“Hong”不变量；其次，为K给出一个逐步的公理基础，以分析它在拥有如此多的Hong不变量方面取得的显著成功。追求第二个目标会导致“本质Kolmogorov”（EK）关系，这种关系族在一个基集上形成一个由包含有序的完整晶格；对角线关系Δ是普遍下界，Kolmogorov关系K是普遍上界——通常在Δ和K之间严格存在许多EK关系，尽管EK关系是K的显著弱化，但它们在r.t Hong不变量中同样成功。反例阐明相似概念之间的关系。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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.