Mathematical Sciences: Investigations on Normal and Paracompact Spaces

虽然正规和准紧空间是经典的和经常使用的拓扑空间类，并且已经被广泛地研究过（事实上，诸如正规摩尔空间和Dowker空间的重大突破正是在这个领域发生的），但它们也是那些仍然存在许多古老和值得关注的问题的类之一。这个项目的目的是在解决这些问题上取得进展，主要是通过创建新的拓扑空间结构。要使用的主要技术起源于M.E.Rudin的一篇论文，其中构造了一个正规但不是集合正规的简单复形的例子。在那篇论文中，Rudin在一组基数c（=实线的基数）上使用大小为2^c的结构的可数子结构来编码正态性。对该技术的改进和改进将用于从文献中攻击七个问题，并获得相关结果。一个典型的这样的问题是是否和哪些正常的，可屏蔽的（=每个开放覆盖有一个sigma不相交的开放细化）空间是准紧的。通常欧几里得空间的以下两个重要性质在数学中经常使用。(1)闭集上的每一个连续函数都可以在整个空间上连续扩展（正态性）。(2)连续的局部结构可以连续地合并为全局结构（准紧性）。50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些性质中的一种或两种；这样的空间分别称为正规空间和准紧空间。七八十年代的重大突破揭示了正常空间和准紧空间的大部分结构。然而，基础研究问题仍未得到解答。这些问题通常是文献中具体而明确的猜想，与以下问题有关：(a)如何在正规空间类中识别准紧空间？(b)两个或两个以上的某种正规空间的积是否也具有这种有用的可拓性？在这个项目的过程中，M.E.Rudin发明的一种技术的改进将用于构建一般的拓扑空间，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. ***