ON SOME RESULTS RELATED TO SOBER SPACES  

作　　者：李庆国 金梦洁 苗化磷 陈思恒 Qingguo LI;Mengjie JIN;Hualin MIAO;Siheng CHEN(School of Mathematics,Hunan University,Changsha 410082,China)

机构地区：[1]School of Mathematics,Hunan University,Changsha 410082,China

出　　处：《Acta Mathematica Scientia》2023年第4期1477-1490,共14页数学物理学报（B辑英文版）

基　　金：the NNSF of China(12231007)。

摘　　要：This paper investigates sober spaces and their related structures from different perspectives.First,we extend the descriptive set theory of second countable sober spaces to first countable sober spaces.We prove that a first countable T_(0) space is sober if and only if it does not contain a∏_(2)^(0)-subspace homeomorphic either to S_(D),the natural number set equipped with the Scott topology,or to S_(1),the natural number set equipped with the cofinite topology,and it does not contain any directed closed subset without maximal elements either.Second,we show that if Y is sober,the function space TOP(X,Y)equipped with the Isbell topology(respectively,Scott topology)may be a non-sober space.Furthermore,we provide a uniform construction to d-spaces and well-filtered spaces via irreducible subset systems introduced in[9];we called this an H-well-filtered space.We obtain that,for a T_(0) space X and an H-well-filtered space Y,the function space TOP(X,Y)equipped with the Isbell topology is H-well-filtered.Going beyond the aforementioned work,we solve several open problems concerning strong d-spaces posed by Xu and Zhao in[11].

关 键 词：sober space descriptive set theory function space strong d-space product space REFLECTION 

分 类 号：O189.11[理学—数学]