The Answer to a Problem Posed by Zhao and Ho

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英文篇名：The Answer to a Problem Posed by Zhao and Ho 作者：Bin ; ZHAO ; Jing ; LU ; Kai ; Yun ; WANG 英文作者：Bin ZHAO;Jing LU;Kai Yun WANG;School of Mathematics and Statistics, Xi'an Jiaotong University;School of Mathematics and Information Science, Shaanxi Normal University; 英文关键词：T0 space;;irreducibly-derived topology;;k-bounded sober space 中文刊名：ACMS 英文刊名：数学学报(英文版) 机构：School of Mathematics and Statistics, Xi'an Jiaotong University;School of Mathematics and Information Science, Shaanxi Normal University; 出版日期：2019-02-26 出版单位：Acta Mathematica Sinica 年：2019 期：v.35 基金：Supported by the National Natural Science Foundation of China(Grant Nos.11531009 and 11871320);; the Fundamental Research Funds for the Central Universities(Grant No.GK201803002) 语种：英文; 页：ACMS201903010 页数：7 CN：03 ISSN：11-2039/O1 分类号：146-152

摘要

Zhao and Ho asked in a recent paper that for each T_0 space X, whether KB(X)(the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Top_κ of the category Top_0 of T_0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Top_κ.
        Zhao and Ho asked in a recent paper that for each T_0 space X, whether KB(X)(the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Top_κ of the category Top_0 of T_0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Top_κ.

引文

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