{"title":"Completely Scott closed set and its applications","authors":"Licong Sun, Bin Pang","doi":"10.1016/j.topol.2025.109283","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we propose a concept of completely Scott closed sets and use it to study links between convex spaces and continuous lattices. Firstly, we take three equivalent approaches to construct a convex space from a continuous lattice. Secondly, we construct an adjunction between the category of convex spaces and the opposite category of continuous lattices via completely Scott closed sets. This adjunction exactly induces the concept of sober convex spaces which gives rise to a categorical duality between them and algebraic lattices. Finally, we prove that completely Scott closed sets form a monad over the category of convex spaces and obtain an isomorphism between the category of sober convex spaces and the Eilenberg–Moore category of this monad.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"365 ","pages":"Article 109283"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-19","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/S0166864125000811","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
In this paper, we propose a concept of completely Scott closed sets and use it to study links between convex spaces and continuous lattices. Firstly, we take three equivalent approaches to construct a convex space from a continuous lattice. Secondly, we construct an adjunction between the category of convex spaces and the opposite category of continuous lattices via completely Scott closed sets. This adjunction exactly induces the concept of sober convex spaces which gives rise to a categorical duality between them and algebraic lattices. Finally, we prove that completely Scott closed sets form a monad over the category of convex spaces and obtain an isomorphism between the category of sober convex spaces and the Eilenberg–Moore category of this monad.
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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