Mathematical Scieces: Topics in Set-Theoretic Topology

9401529 Gruenage是集合论拓扑学的首席研究员，他将继续研究集合论拓扑学中的几个未决问题。焦点将集中在Arangel‘skii和Tall这两个经典问题上：正规的局部紧的亚紧空间是仿紧的吗？由于首席研究员最近取得的进展，这个问题的解决方案可能近在咫尺。另一组问题围绕着乘积中的Lindelof性质的失败，并包括Michael长期存在的关于ZFC中是否存在其与完全可分度量空间的乘积不是Lindelof的问题。然而，第三组问题涉及PI最近对紧Hausdorff空间的闭划分上的Watson、Dow和Porter问题的解决方案的扩展。集合论和一般拓扑学是基本的数学学科，它们有着共同的世纪之交根源。集合论提供了一个框架，所有的数学都可以基于这个框架。拓扑学的研究对象包括熟悉的空间，如平面和普通三维欧几里德空间，以及它们的子集，以及更抽象的空间。歌德尔和科恩的开创性工作导致人们认识到，一些长期存在的拓扑问题不能仅假设集合论的标准公理来解决。确定拓扑命题在这个意义上是否独立于标准公理是很有启发性的，如果是这样的话，需要什么进一步的公理来解决它们。这个项目就涉及到这样的问题，涉及基本的拓扑性质和构造，包括乘积空间、划分和覆盖的精化。要研究的问题包括那些可以追溯到20世纪60年代S的问题，到目前为止，许多专家已经研究过这些问题，但只取得了部分成功。首席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方案现已指日可待。他们的解决方案中涉及的新技术难题很可能适用于各种各样的其他问题。***

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. ***