仿拓扑群中的三空间问题

The combination of topology and algebra has a long-standing history. From the intersection and integration of topology and algebra, there generates the theory of the topological algebra, which has become one of the active front research field in the study of general topology. In recent years, members of the project team have opened up the research approach to the theory of generalized metric spaces in Topology Algebra. Focusing on several open problems about three spaces in (para-)topological groups raised in the 2008 edition of monograph "Topological Groups and Related Structures", the project team will explore approaches of solving Arhangelsk'iǐ and Tkachenko's problems by using methods of set theory, topology, algebra and statistics, and will form gradually a research base of Topological Algebras in China. The project team goals at to publish more than 30 papers in journals with international influence and one monograph, and to make our research work being on the frontier of the field.

拓扑与代数的结合由来已久。由一般拓扑学与抽象代数学融合而发展起来的拓扑代数已渐成活跃的前沿研究领域之一。近年来本项目组成员开辟了拓扑代数中广义度量空间理论的研究途径。本项目将围绕2008年出版的《Topological Groups and Related Structures》等中关于（仿）拓扑群的三空间的若干问题，借助集合论、拓扑学、代数学与统计学中的成功方法，寻求 Arhangelskiǐ及Tkachenko问题的解决，逐步形成国内拓扑代数的研究基地，在有国际影响的刊物上发表论文30篇以上，出版著作1部，使我们的若干研究工作能处于国际前沿。

拓扑代数是近年来一般拓扑学的活跃研究领域，其三空间问题成为连结拓扑与代数的纽带之一。本项目首次提出并研究仿（拟）拓扑群的三空间问题，探讨广义度量空间上自由（阿贝尔，仿）拓扑群的弱第一可数性，出版著作3种、发表论文74篇，其中在《Topology and its Applications》上发表论文24篇，举办5场学术会议；主要结果是证明局部紧性、连通性、紧集的第一可数性等都是仿拓扑群的三空间性质，第一可数性是阿贝尔仿拓扑群的三空间性质，刻画广义度量空间上自由（阿贝尔，仿）拓扑群的可数tightness、k空间性质、k_R空间性质和Fréchet性质等，并且回答A.V. Arhangel’skii和M. Tkachenko在专著《Topological Groups and Related Structures》中提出的若干问题。在一般拓扑学及其应用方面的探索将促进集论、拓扑、代数及不确定研究的融合与发展。