稠可分群及其在d独立性的应用  

Dense-separable Groups and Its Applications in d-independence

作　　者：林福财 吴琪韵 刘川[2] Fucai Lin;Qiyun Wu;Chuan Liu(School of Mathematics and Statistics,Minnan Normal University,Zhangzhou 363000,P.R.China;Department of Mathematics,Ohio University Zanesville Campus,Zanesville,OH 43701,USA)

机构地区：[1]闽南师范大学数学与统计学院,漳州363000 [2]俄亥俄大学赞斯维尔校区数学系,美国俄亥俄州43701

出　　处：《数学学报(中文版)》2025年第2期379-396,共18页Acta Mathematica Sinica:Chinese Series

基　　金：国家自然科学基金(11571158);福建省自然科学基金重点项目(2024J02022)。

摘　　要：拓扑空间称为稠可分的,若每个稠子集都是可分的.因此,每个稠可分空间是可分的.本文主要讨论稠可分拓扑群的一些基本性质,证明了可分的且具有可数tightness的空间是稠可分的,并且给出例子说明稠可分的拓扑群不一定是遗传可分的;然后,证明了Hausdorff局部紧群是稠可分当且仅当它是可度量化的.此外,本文研究了稠子群可分的拓扑性质,证明了每一交换的、局部紧的拓扑群是稠子群可分的当且仅当它是稠可分的,当且仅当它是可度量化的.最后,本文还讨论稠可分在拓扑群及其相关结构的d独立方面的一些应用,主要证明了如下结果:(1)每一正则、稠子群可分且无挠秩不小于连续基数的交换半拓扑群是d独立的.(2)对每一正则、有界的交换仿拓扑群G,若G是稠子群可分且|G|>1,则G是d独立的当且仅当G是M群,当且仅当G的每一非平凡准素分支G_(p)是d独立的;并运用该结果,证明了可分度量化的几乎无挠仿拓扑交换群G满足|G|=c是d独立的.(3)每一具有非平凡连通分支、MAP且稠子群可分的交换群是d独立的.A topological space is called dense-separable if each dense subset of it is separable.Therefore,each dense-separable space is separable.This paper is devoted to establishing some basic properties of dense-separable topological groups.We prove that each separable space with a countable tightness is dense-separable,and give a dense-separable topological group which is not hereditarily separable.We also prove that,for a Hausdorff locally compact group,it is locally dense-separable iff it is metrizable.Moreover,we study dense-subgroup-separable topological groups.We prove that,for each locally compact abelian group,it is dense-subgroup-separable iff it is denseseparable iff it is metrizable.Finally,we discuss some applications in d-independent topological groups and related structures.We prove that each regular dense-subgroupseparable abelian semitopological group with r_(0)(G)≥c is d-independent.We also prove that,for each regular dense-subgroup-separable bounded paratopological abelian group G with|G|>1,it is d-independent iff it is a nontrivial M-group iff each nontrivial primary component G_(p)of G is d-independent.Applying this result,we prove that a separable metrizable almost torsion-free paratopological abelian group G with|G|=c is d-independent.Further,we prove that each dense-subgroup-separable MAP abelian group with a nontrivial connected component is also d-independent.

关 键 词：稠可分 稠子群可分 d独立 交换群 可分空间 度量化 半拓扑群 紧交换群 

分 类 号：O189.2[理学—数学]