Investigations on Normal and Paracompact Spaces

题目：正规空间和准紧空间的研究摘要：本课题涉及通过初等子模型利用对角化构造拓扑空间技术的应用。将这种技术应用于正规到拟紧空间的基本类是由M.E.Rudin首创的。在之前的拨款期间，首席研究员对其进行了改进和扩展，以获得该领域未解决的一些问题的答案。具体的改进包括完全邻域技术、反射技巧以及新的应用程序。这样的构建是费力的，因此所针对的问题需要是(a)众所周知的，最好是古老的，被认为是基本的；(b)没有已知的特别解决办法；(c)适合于某种“自然的”归纳构造。这类问题的例子在从正规到准紧的空间中比比皆是。预计其应用将扩展到拓扑学、分析学和无穷组合学等其他领域。集合论拓扑研究的目的是利用集合论和模型论的方法来发现抽象拓扑空间的性质。这样的研究在具体的数学学科的背景下，建立了处理无限对象和过程的具体技术。预计这些技术将适用于数学的其他领域，并与集合论和模型论的相互作用将得到丰富。

Proposal: DMS9977099PI: Zoltan BaloghTitle: Investigations on normal and paracompact spacesAbstract: This project involves applications of the technique of constructing topological spaces using diagonalizations through elementary submodels. Applications of this technique to the fundamental classes of normal to paracompact spaces were pioneered by M.E.Rudin. It has been refined and expanded by the Principal Investigator during the previous grant period to get answers to some of the questions left undecided in this area.Specific improvements include the technique of complete neighborhoods, reflection tricks as well as new applications. Such constructions are laborious, and so the problems targeted need to be (a) well-known, preferably old and considered basic; (b) have no known ad hoc solutions; (c) lend themselves to some "natural" inductive construction. Examples of such problems are abundant in the classes of spaces ranging from normal to paracompact. It is expected that the applications will reach into other areas of topology, analysis and infinite combinatorics. Research in set-theoretic topology is aimed at finding out the properties of abstract topological spaces using methods of set theory and model theory. Such research builds, in the setting of a concrete mathematical discipline, concrete techniques of handling infinite objects and processes. It is expected that the techniques will be applicable in other areas of mathematics and interactions with set theory and model theory will be enriched.