【作者】 王汉锋；

【导师】 贺伟；

【作者基本信息】 南京师范大学 ， 基础数学， 2014， 博士

【摘要】 本文研究拓扑空间(包括拓扑群,半拓扑群,仿拓扑群以及齐性空间)在紧化空间中的剩余问题,主要讨论剩余的广义度量化性质与空间自身的广义度量化性质之间的关系.此外,文章还对K-空间及拓扑群同胚等问题进行了研究.第1章给出了本文研究的历史背景、主要结果及预备知识.在第1节,我们主要回顾了国内外在紧化剩余方面的研究历史、现状,给出了本文的主要结果.第2节给出了文中用到的一些基本概念及一些相关命题.第2章研究了一般拓扑空间及仿拓扑群的紧化剩余,改进和推广了A.VArhangel’skii的两个结果,得到了如下结果：(a)设bG是非局部紧仿拓扑群G的一个紧化空间,则Y=bG\G是局部p-空间当且仅当下列两个条件中的一个成立：(1)G是Lindelof p-空间；(2)G是σ-紧空间.(b)设bX是无处局部紧的完全正则空间X的一个紧化空间,如果满足：(1)X具有局部Gδ-对角线；(2)Y=bX\X是仿紧p-空间,则X是可分的度量空间.第3章讨论了半拓扑群与Rectifiable空间的紧化剩余,改进和推广了A.VArhangel’skii及刘川的几个结果,主要结果如下：(c)设bG是非局部紧半拓扑群G的一个紧化空间,如果Y=bG\G具有局部点可数基,则G与bG都具有可数基.(d)设bX是非局部紧的局部仿紧rectifiable空间X的一个紧化空间,如果Y=bX\X具有局部Gδ-对角线,则X与Y都具有可数基.第4章讨论了K-空间及CS-空间,肯定地回答了由V.IMalykhin与G.Tironi在文献[62]中提出的一个公开问题：(e)紧的K-空间具有可数紧度吗?此外本章还讨论了K-拓扑群及CS-空间的映射性质.第5章讨论了拓扑群的同胚问题,解决了A.V.Arhangel’skii与M.Tkachenko在文献[33]中提出的两个公开问题：(f)每一个连通的拓扑群都是w-narrow拓扑群吗?(g)设e是零维拓扑群G的单位元,子空间G、｛e｝一定与某个拓扑群同胚吗?更多还原

【Abstract】 In this thesis, remainders of topological spaces (includes paratopological groups, semitopological groups and homogeneous spaces) in some compactifications are stud-ied. We mainly investigate the relationship between the generalized metrizability properties of remainders and the generalized metrizability properties of the space it-self. In addition, we also discuss K-spaces and the homeomorphisms of topological groups.In Chapter1, some background, main results and preliminaries are given. In Section1, we recall the research progress of remainders of topological spaces at home and aboard, and offer the main results in this thesis. In Section2, we introduce some preliminary knowledge, including some important concepts and related propositions.In Chapter2, we investigate remainders of general topological spaces and paratopological groups, two theorems by A.V. Arhangel’skii are improved. The fol-lowing two main results are established:(a) If bG is a compactification of a non-locally compact paratopological group G, then Y=bG\G is locally a p-space if and only if at least one of the following conditions holds:(1) G is a Lindelof p-space;(2) G is σ-compact.(b) If bX is a compactification of a nowhere locally compact space X with locally a Gδ-diagonal such that Y=bX\X is a paracompact p-space, then X is separable and metrizable.In Chapter3, we study remainders of semitopological groups and rectifiable spaces, several results given by A.V. Arhangel’skii and Chuan Liu are improved and generalized. Main results are as follows:(c) If a non-locally compact semitopological group G has a compatification bG such that Y=bG\G has locally a point-countable base, then both G and bG are separable and metrizable.(d) If a non-locally compact locally paracompact rectifiable space X has a com-pactification bX satisfying that Y=bX\X has locally a Gδ-diagonal, then X and Y are both have countable bases.In Chapter4, K-spaces and CS-spaces are discussed. An open problem posed by V.I. Malykhin and G. Tironi in [62] is answered firmly: (e) Must a compact K-space have countable tightness?We also investigate K-topological groups and the mapping properties of CS-spaces.In Chapter5, we investigate the homeomorphisms of topological groups, and two open problems posed by A.V. Arhangel’skii and M. Tkachenko in [33] are solved:(f) Is every connected topological group homeomorphic to an w-narrow topolog-ical group?(g) For a zero-dimensional topological group G with neutral element e, must X=G\{e} be homeomorphic to a topological group?更多还原