仿拓扑群与半拓扑群的一些性质

Algebra and topology, the two fundamental domains of mathematics, play complementary roles. Because of the difference in nature, algebra and topology have a strong tendency to develop independently, not in direct contact with each other. However, in applications, in higher level domains of mathematics such as functional analysis, topology and algebra come in contacts mostly naturally. Many of the most important objects of mathematics represent a blend of algebraic and topological structures. .The combining of algebra and topology has raised many interesting problems. One of the general questions in topological algebra is how the topological properties depend on the underlying algebraic structure. The theory of topological group is core subject of topological algebra. Results on topological groups are followed by a discussion of other structures of topological algebra, such as semitopological groups and paratopological groups. It is well-known that every first-countable topological group is metrizable, but it is not true for the case of paratopological group. For example, the Sorgenfrey line is a paratopological group which is not metrizable. We know that every topological group is completely regular. It is still unknown if every regular paratopological group is completely regular. It is natural to study what properties of topological groups can be also possessed by paratopological groups or semitopological groups. .Recently we have been studying topological spaces with algebraic structures,especially properties of paratopological groups and semitopological groups. We have answered 7 questions asked by the famous topologist Arhangelskii and other active topologists. For example, we found a paratopological group which is a Moore space but not a metrizable space. It answered an open problem by Ravsky in 2001. We proved that every first-countable narrow paratopological group is separable. We proved every locally pseudocompact paratopological group is a topological group. .The purpose of this research project is to continue the study of paratopological groups or semitopological groups. It mainly consists of: relationship of paratopological groups or semitopological groups and topological groups; generalized metrization problems of paratopological groups or semitopological groups; base covering properties of topological groups. We have gotten some useful results on the study of metrization problems and base covering properties, which will promote the development of this project. The study of this project will combine theories and methods of general topology, set theory, algebra and analysis. It will be a very important research topic.

拓扑代数是指赋予代数结构的拓扑空间。拓扑群理论是拓扑代数的核心课题，拓扑群有很多良好的拓扑性质，如第一可数的拓扑群是可度量化的。拓扑群的研究推动了更广义的拓扑代数理论的发展，如仿拓扑群和半拓扑群。拓扑群的许多重要性质能否推广到仿拓扑群或半拓扑群上是拓扑学家们关心的问题。最近我们一直在研究相关问题，并回答了Arhangel'skii 等提出的7个问题，如：我们找到了一个仿拓扑群是Moore空间但不是可度量空间；证明了每一个局部伪紧的仿拓扑群是拓扑群；证明了每一个第一可数的 narrow仿拓扑群是可分的。本项目将继续研究仿拓扑群和半拓扑群的拓扑性质，主要讨论以下几个方面的问题：仿拓扑群或半拓扑群与拓扑群及映射的关系；仿拓扑群或半拓扑群的广义度量性质；仿拓扑群的基覆盖性质等。我们将结合一般拓扑学、公理集合论、代数与分析的理论和方法，对这样的具有代数结构的拓扑空间进行深入的研究。

拓扑群是具有拓扑结构的群。拓扑群的研究推动了更广义的拓扑代数理论的发展，如仿拓扑群，半拓扑群，拟拓扑群等。近十几年，拓扑代数逐渐发展成一般拓扑学非常活跃的研究领域。2008年A.V.Arhangelskii和M.Tkachenko出版了他们的拓扑代数方面的专著《Topological Groups and Related Structrues》，总结了国际上关于拓扑代数领域的研究成果，提出了大量的公开问题。. 本项目主要研究仿拓扑群、半拓扑群和拟拓扑群的拓扑性质，重点讨论以下几个方面的问题：仿拓扑群、半拓扑群及拟拓扑群与拓扑群的关系；仿拓扑群、半拓扑群及拟拓扑群的广义度量性质。我们的研究回答了《Topological Groups and Related Structrues》中的多个问题。另外我们对拓扑空间的覆盖性质及Mrowka空间的对角线性质进行了深入的研究，得到了丰富的结果。. 项目进展顺利，达到了预期目标。已发表或录用论文12篇，其中10篇为SCI,出版专著1部，参加国际拓扑会议3次，协助组织国际拓扑会议1次，邀请多位国际著名拓扑学家访问中国。