Investigations on Normal and Paracompact Spaces

正规空间和仿紧空间的研究

• 批准号: 9977099
• 负责人: Dennis Burke
• 金额: $ 7.43万
• 依托单位: Miami University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9977099&HistoricalAwards=false
• 关键词: Investigations; Normal; Paracompact; Spaces

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.

提议：DMS9977099PI：Zoltan Balogh标题：关于正规和仿紧空间的研究摘要：这个项目涉及通过基本子模型的对角化来构造拓扑空间的技术的应用。M.E.Rudin率先将这一技巧应用于正规到仿紧空间的基本类。在之前的拨款期间，首席调查员对它进行了改进和扩展，以获得这一领域一些悬而未决的问题的答案。具体的改进包括完整邻域的技术、反思技巧以及新的应用。这种构造是费力的，因此目标问题需要(A)众所周知，最好是陈旧的，并被认为是基本的；(B)没有已知的特别解决方案；(C)适合一些“自然”的归纳构造。这类问题的例子在从正规到仿紧的空间类中比比皆是。预计这些应用将延伸到拓扑学、分析和无限组合学的其他领域。集合论拓扑学的研究目的是利用集合论和模型论的方法找出抽象拓扑空间的性质。这种研究在一个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。预计这些技术将应用于数学的其他领域，并将丰富与集合论和模型理论的互动。