Problems in Set-Theoretic Topology

首席研究员将继续他的研究在集理论拓扑几个开放的问题。 首先，他将确定他最近的解决方案（与P. Koszmider）的著名的Arhangel'skiii-Tall问题的一些修改可以应用于其他长期存在的开放问题，这些问题涉及局部好空间中的正规性和仿紧性。 这些问题包括M. E。Rudin的问题是否每个正规流形必须是集体Hausdorff，和S。沃森关于ZFC中完全正规局部紧非仿紧空间存在性的问题。 第二，两个问题有关的封闭映射将被调查，第一个是著名的公开问题的metalindelof财产的保护下封闭或完善的地图，和其他问题的存在性不可约限制的封闭映射。 最后，一组其他问题有关最近的工作的主要研究者包括问题的特征的Baire财产的功能空间的紧开拓扑，存在一个完全正规连通的度量空间没有紧开集，和问题的行为的Lindelof财产的产品空间。 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。 K. 30年代末的哥德尔和60年代的P.科恩很快使人们认识到，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。 主要研究者的研究基本上是在这两个领域之间的有趣的相互作用的范围内，特别侧重于为它们的拓扑意义选择的问题，但在那里集合论的知识可能是有用的，在他们的解决方案。 这些问题涉及基本的拓扑性质和构造-正规性、覆盖性质、乘积空间、函数空间等。它影响了一般拓扑学领域的很大一部分，并且在数学的其他领域也有应用。 解决这些问题所涉及的新技术很可能适用于各种各样的其他问题。

The principal investigator will continue his research on several open problems in set-theoretic topology. First, he will determine if some modifications of his recent solution (with P. Koszmider) of the well-known Arhangel'skii-Tall problem can be applied to other long-standing open problems concerning normality and paracompactness in locally nice spaces. These problems include M.E. Rudin's question of whether or not every normal manifold must be collectionwise-Hausdorff, and S. Watson's question about the existence in ZFC of a perfectly normal locally compact non-paracompact space. Second, two problems concerning closed mappings will be investigated, the first being the well-known open question about the preservation of the metalindelof property under closed or perfect maps, and the other a question about the existence of irreducible restrictions of closed mappings. Finally, a set of other problems related to recent work of the principal investigator includes the question of the characterization of the Baire property in function spaces with the compact-open topology, of the existence of a completely regular connectification of metrizable spaces with no compact open sets, and problems about the behavior of the Lindelof property in product 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. The pioneering set-theoretical work of K. Goedel in the late 30's and P. Cohen in the 60's soon led to the realization that some long-standing topological questions could not be solved assuming only the standard axioms of set-theory. The principal investigator's research lies substantially within the scope of this interesting interaction between the two fields, focusing especia lly on problems selected for their topological significance, but where knowledge of set-theory is likely to be useful in their solution. The problems involve basic topological properties and constructions-normality, covering properties, product spaces, function spaces, etc.-which impact a large portion of the field of general topology, and have applications in other areas of mathematics as well. The new techniques involved in the solution of these problems are likely to be applicable to a wide variety of other problems.