自由(仿)拓扑群中广义度量性质的刻画

From the intersection and integration of topology and algebra, there generates the theory of topological algebra, the frontier in the study of general topology, where the theory of free topological groups is one of the most dynamic research objects. Focusing on the open problems concerning about the generalized metric properties of free topological group and free paratopological groups in monographs like Topological Groups and Related Structures, Recent Progress in General Topology Ⅱ and Recent Progress in General Topology Ⅲ, we hope that we can obtain some characterizations of generalized metric properties in free topological groups and the imbedded problem in the free paratopological groups, which can make us easily understand the intrinsic relations. On the basis of what have been studied and achieved, the project group will research collaboratively on the Arhangel’skii’s problems, Nickolas’s problems and Tkachenko’s problems, and apply the generalized metric methods in general topology and the skills of the operators in algebra to studying the generalized metrizabilities and the theory of free topological groups, with the hope of publishing about 30 papers in influential journals.

源于拓扑和代数的交叉与融合而产生的拓扑代数已成为一般拓扑学的前沿研究方向，其中自由拓扑群理论是最有活力的研究对象之一。围绕《Topological Groups and Related Structures》、 《Recent Progress in General Topology Ⅱ》和 《Recent Progress in General Topology Ⅲ》等中关于自由(仿)拓扑群的广义度量性质的公开问题，建立一些自由(仿)拓扑群上经典的广义度量性质的刻画、自由仿拓扑群嵌入问题，阐述拓扑空间的广义度量空间与其上的自由拓扑群更深刻之内在属性。我们将在已取得良好工作的基础上，对Arhangel’skii问题、Nickolas问题、Tkachenko问题等进行协作攻关，把一般拓扑学中的广义度量方法和代数学中的运算技巧用于自由拓扑群理论的研究，争取在有影响的杂志上发表论文30篇左右。

由拓扑和代数的交叉与融合而产生的拓扑代数已成为一般拓扑学的前沿研究方向，其中自由拓扑群理论是最有活力的研究对象之一。围绕《Topological Groups and Related Structures》、 《Recent Progress in General Topology Ⅱ》和 《Recent Progress in General Topology Ⅲ》等中关于自由(仿)拓扑群的广义度量性质的公开问题，建立了一些自由(仿)拓扑群上经典的广义度量性质的刻画及其相关应用，阐述拓扑空间的广义度量空间与其上的自由拓扑群更深刻之内在属性。我们对Arhangel’skii问题、Yamada问题等进行协作攻关，把一般拓扑学中的广义度量方法和代数学中的运算技巧用于自由拓扑群理论的研究，在有影响的期刊或会议上发表论文43篇。