Mathematical Sciences: Investigations on Normal and Paracompact Spaces

9623391 Balogh虽然正规和仿紧空间是一类经典的和经常使用的拓扑空间，并且已经被广泛地研究过(事实上，关于正规Moore空间和Dowker空间的重大突破正是在这个领域发生的)，它们也是一些古老而古老的问题仍然悬而未决的类别之一。这个项目的目的是在解决其中一些问题上取得进展，主要是通过创建新的拓扑空间结构来实现。所使用的主要技巧起源于M.E.Rudin的一篇论文，其中构造了一个正规但不是集合正规单纯形的例子。在这篇文章中，Rudin使用一组基数c(=实直线的基数)上大小为2^c的结构的可数子结构来编码正规性。对这一技术的改进和改进将被用来解决文献中的七个问题，并获得相关结果。一个典型的这样的问题是，正规的、可屏蔽的(=每个开覆盖有一个sigma不相交的开加细)空间是否是仿紧的，以及哪一个是仿紧的。通常的欧几里德空间的下列两个重要性质在数学中经常被使用。(1)闭集上的每个连续函数都可以在整个空间上连续扩张(正规性)。(2)连续的局部结构可以连续地融合成全局结构(仿紧性)。五十年代早期及之前的研究表明，数学应用中使用的许多更一般类型的空间也具有这两种性质中的一种或两种；具有这两种性质的空间分别称为正规空间和仿紧空间。七八十年代的重大突破揭示了正常和超紧凑空间的大部分结构。然而，基础研究的问题仍然没有得到回答。这样的问题在文献中是典型的具体的、明确的猜想，与以下问题有关：(A)人们如何识别正规空间类中的仿紧空间？(B)某一类的两个或多个正规空间的乘积是否也具有这种有用的扩张性质？在这个项目的过程中，由M.E.Rudin提出的一种技术的改进将被用来建立一般的拓扑空间，目的是解决上面提到的一些猜想。***

9623391 Balogh Although normal and paracompact spaces are classical and frequently used classes of topological spaces, and have been extensively investigated (indeed, major breakthroughs such as those on normal Moore spaces and Dowker spaces took place exactly in this area), they are also among those classes in which a number of old and venerable problems remain open. The aim of this project is to make progress in settling some of these problems, mainly by creating new constructions of topological spaces. The principal technique to be used has its origin in a paper of M.E.Rudin, where an example of a normal but not collectionwise normal simplicial complex is constructed. In that paper Rudin uses countable substructures of a structure of size 2^c on a set of cardinality c (= the cardinality of the real line) to code normality. Refinements of and improvements on this technique will be used to attack seven problems from the literature as well as to obtain related results. A typical such problem is whether and which normal, screenable (= every open cover has a sigma-disjoint open refinement) spaces are paracompact. The following two important properties of the usual Euclidean space are often used in mathematics. (1) Every continuous function on a closed set can be continuously extended over the entire space (normality). (2) Continuous local structures can be continuously amalgamated into global ones (paracompactness). Research during and before the early fifties showed that many of the more general types of spaces used in mathematical applications also have one or both of these properties; those that do are called, respectively, normal and paracompact spaces. Major breakthroughs in the seventies and eighties uncovered much of the structure of normal and paracompact spaces. However, basic research questions remain unanswered. Such questions are typically concrete, explicit conjectures in the literature that have to do with problems such as these: (a ) How does one recognize paracompact spaces in the class of normal spaces? (b) Does the product of two or more normal spaces of a certain kind also have this useful extension property? During the course of this project, refinements of a technique originated by M.E.Rudin will be used to build general topological spaces, with the goal of settling some of the conjectures mentioned above. ***