Mathematical Scieces: Topics in Set-Theoretic Topology

数学科学：集合论拓扑主题

• 批准号: 9401529
• 负责人: Gary Gruenhage
• 金额: $ 5.22万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-08-01 至 1997-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401529&HistoricalAwards=false
• 关键词: Mathematical; Scieces; Topics; Set; Theoretic

9401529 Gruenhage The principal investigator will continue his research on several open questions in set-theoretic topology. The focus will be on the now classical problem of Arhangel'skii and Tall: Are normal locally compact metacompact spaces paracompact? Because of recent progress by the principal investigator, a solution to this problem may be near. Another set of problems revolves around the failure of the Lindelof property in products, and includes Michael's long-standing question concerning the existence in ZFC of of a Lindelof space whose product with a complete separable metric space is not Lindelof. Yet a third set of problems involves extensions of the PI's recent solutions to problems of Watson, and Dow and Porter, on closed partitions of compact Hausdorff spaces. Set-theory and general topology are fundamental mathematical disciplines which have common turn-of-the-century roots. Set- theory provides a framework upon which all of mathematics can be based. Objects of study in topology include familiar spaces such as the plane and ordinary three-dimensional Euclidean space, and their subsets, as well as much more abstract spaces. Pioneering work of Goedel and Cohen led to the realization that some long- standing topological questions could not be solved assuming only the standard axioms of set-theory. It is illuminating to determine whether or not topological propositions are independent of the standard axioms in this sense, and if so, what further axioms are necessary to settle them. This project involves just such questions, concerning basic topological properties and constructions, including product spaces, partitions, and refinements of coverings. Questions to be studied include ones which date back to the 1960's and have been worked on by many experts with only partial success so far. Significant recent progress by the principal investigator indicates that a complete solution to some of these may now be at hand. The new tech niques involved in their solution are likely to be applicable to a wide variety of other problems. ***

小行星9401529 首席研究员将继续他的研究在集理论拓扑的几个开放的问题。 重点将放在现在的经典问题的Arhangel'skii和高：正常的局部紧亚紧空间仿紧？ 由于主要研究者最近取得的进展，这个问题的解决方案可能即将到来。 另一组问题围绕着失败的林德洛夫财产的产品，包括迈克尔的长期存在的问题，关于存在ZFC的林德洛夫空间的产品与一个完整的可分度量空间是不是林德洛夫。 然而第三组问题涉及扩展PI的最近解决方案的问题的沃森，道和波特，封闭分区的紧凑型豪斯多夫空间。 集合论和一般拓扑学是数学的基本学科，它们有着共同的世纪之交的根源。 集合论提供了一个所有数学都可以基于的框架. 拓扑学的研究对象包括熟悉的空间，如平面和普通的三维欧氏空间，以及它们的子集，以及更抽象的空间。 哥德尔和科恩的开创性工作使人们认识到，一些长期存在的拓扑问题不能仅仅假设集合论的标准公理来解决。 在这个意义上，确定拓扑命题是否独立于标准公理，以及如果是的话，需要什么进一步的公理来解决它们，这是很有启发性的。 这个项目涉及的正是这样的问题，关于基本的拓扑性质和结构，包括产品空间，分区，覆盖和细化。 要研究的问题包括那些可以追溯到20世纪60年代的问题，许多专家一直在研究这些问题，但迄今为止只取得了部分成功。 主要调查员最近取得的重大进展表明，其中一些问题的彻底解决办法现在可能就在眼前。 他们的解决方案所涉及的新技术可能适用于各种各样的其他问题。 ***