拟拓扑群中若干问题的研究

Topological algebra is an organic integration of topology and algebra. As a generalization of topological groups, quasitopological groups not only inherit some properties of topological groups, but also differ greatly from them. The study of quasitopological groups has not only theoretical value, but also can be applied to other branches. This project focuses on the generalized metric properties and cardinal invariants of quasitopological groups and the applications of quasitopological groups in fundamental groups. According to some open problems about quasitopological groups posed by A.Arhangel'skiĭ, M.Tkachenko, A.Tomita, J.Brazas and P.Fabel, we will focus attention on the following three open problems: (1) Let G be a Hausdorff quasitopological group such that every compact subspace of G is first countable. Is every compact subspace of G metrizable? (2) Does every (Hausdorff or regular)σ-compact quasitopological group have countable cellularity? (3) Must the quasitopological fundamental group of X be normal for every compact metric space X? The intensive study of these problems will provide new methods and tools for the study of quasitopological groups, and enrich the theory and applications of quasitopological groups.

拓扑代数是拓扑与代数的有机融合。拟拓扑群作为拓扑群的推广,既继承了拓扑群的某些性质,又与拓扑群有很大的不同。对拟拓扑群的研究不仅具有理论价值,且能应用于其他分支。本项目围绕由A.Arhangel'skiĭ、M.Tkachenko、A.Tomita、J.Brazas与P.Fabel提出的3个公开问题:(1)设G是Hausdorff的拟拓扑群且G的每个紧子空间是第一可数的,那么G的每个紧子空间是可度量的吗?(2)是否每一(Hausdorff或正则的)σ紧拟拓扑群具有可数胞腔度?(3)若X是紧度量空间,那么X的拟拓扑基本群是正规的吗？着重研究拟拓扑群的广义度量性质、基数不变量及拟拓扑群在基本群中的应用。对这些问题的深入研究将为拟拓扑群的相关研究提供新的思路和方法,并进一步丰富拟拓扑群理论及其应用。

拓扑群是拓扑代数的一个主要研究方向,在诸多数学分支中具有丰富的应用.拟拓扑群作为拓扑群的重要推广之一,具有重要的理论和应用价值.本项目主要研究拟拓扑群的广义度量性质、基数不变量及拟拓扑群在基本群中的应用. 项目所取得的成果彻底解决了2个关于拟拓扑群基数不变量的公开问题;部分解决了8个拟拓扑群广义度量性质与基数不变量相关的公开问题;部分解决了1个拟拓扑群cross-complementary相关的公开问题. 研究了拟拓扑群的反射,利用空间与映射的理论建立的拟拓扑群及其反射之间拓扑性质的联系.这些成果不但丰富了拟拓扑群理论,所采用的方法和手段也为拓扑代数理论的研究提供了新的途径,在有影响的期刊上发表标注项目编号的论文8篇.