Problems in Set-Theoretic Topology

首席研究员将继续对集合论拓扑中的几个开放问题进行研究。首先，他将确定他最近（与P. Koszmider）解决的著名的Arhangel'ski - tall问题的一些修改是否可以应用于其他长期存在的关于局部良好空间中的正态性和副紧性的开放问题。这些问题包括M.E. Rudin关于是否所有正规流形都必须是集合- hausdorff的问题，以及S. Watson关于在ZFC中存在一个完全正规的局部紧非拟紧空间的问题。其次，我们将研究两个关于闭映射的问题，第一个是关于闭映射或完全映射下元indedelof性质的保存问题，另一个是关于闭映射的不可约限制的存在性问题。最后，本文还讨论了一系列与本研究相关的其他问题，包括紧开拓扑函数空间中Baire性质的刻画问题，无紧开集的可度量空间的完全正则连通的存在性问题，以及乘积空间中Lindelof性质的行为问题。集合论和一般拓扑学是基本的数学学科，它们具有共同的世纪之交根源。集合论提供了一个框架，所有的数学都可以建立在这个框架之上。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几里得空间及其子集，以及更为抽象的空间。哥德尔（K. Goedel）在30年代末和科恩（P. Cohen）在60年代的开创性的集合论工作，很快使人们认识到，一些长期存在的拓扑问题不能仅用集合论的标准公理来解决。首席研究员的研究基本上是在这两个领域之间有趣的相互作用的范围内，特别关注因其拓扑意义而选择的问题，但集合论的知识可能在其解决方案中有用。这些问题涉及基本的拓扑性质和构造——正态性、覆盖性质、积空间、函数空间等，它们影响了一般拓扑领域的很大一部分，并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.