Problems related to D-spaces.

与 D 空间相关的问题。

• 批准号: 238944-2012
• 负责人: Szeptycki, Paul
• 金额: $ 0.87万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635493
• 关键词: Problems; related; spaces.

This proposal is devoted to research at the intersection of two fields of pure mathematics, Topology and Set Theory. Topological ideas are central to areas of mathematics where notions of distance play a role, and is concerned with those properties of abstract geometric objects unaffected by changes or deformations that preserve nearness relationships. This is a beautiful area of mathematics with great power to simplify and clarify ideas from seemingly divergent areas. For example, fundamental notions from Calculus are simple consequences of topological ideas. Set Theory, on the other hand, forms the foundations for mathematics as a whole. A typical foundational question from Set Theory is, "what are the basic assumptions (axioms) that we use and that are needed in mathematics?" A consequence of Godel's Incompleteness Theorem is that for any acceptable axiom system for mathematics there are mathematical statements that can neither be proven nor refuted from the system. Knowing that a particular mathematical conjecture is independent in this sense is not only of obvious intrinsic interest (knowing a conjecture is independent means we can stop looking for a proof of it), but it also establishes a clear connection to the area of Set Theory. Establishing connections between apparently disparate mathematical fields has always produced interesting and important mathematics. This research proposal is focused on problems concerning the fundamental structure of topological spaces where deep set theoretic connections have either already been established or where we believe they exist. The results of this proposal should lead to the development of new and important ideas in both set theory and general topology, and we hope will find applications to other areas of mathematics. Questions formulated in this proposal are of great interest to experts in the field not only in Canada but also internationally. In addition to expanding the frontier of knowledge regarding topological spaces, this research proposal will continue to promote Canada as a centre of research in pure mathematics.

该建议致力于在两个纯数学，拓扑和集合理论的两个领域的交集中进行研究。拓扑思想对于距离概念发挥作用的数学领域至关重要，并且与不受保留近乎关系关系的变化或变形影响的抽象几何对象的特性有关。这是一个美丽的数学领域，具有很大的力量，可以简化和阐明看似不同的领域的想法。例如，微积分的基本观念是拓扑思想的简单后果。另一方面，设定理论构成了整个数学的基础。集合理论的一个典型基础问题是：“我们使用的基本假设（公理）是什么？ Godel不完整定理的结果是，对于任何可接受的公理系统用于数学，都有数学陈述既不能被证明也无法从系统中证明也无法被驳斥。知道特定的数学猜想在这个意义上是独立的，不仅具有明显的内在兴趣（知道猜想是独立的，这意味着我们可以停止寻找它的证据），而且还与设定理论的领域建立了明确的联系。在明显不同的数学领域之间建立联系一直产生有趣而重要的数学。该研究建议集中在有关拓扑空间的基本结构的问题上，在这些问题上已经建立了深度的理论联系，或者我们认为它们存在。该提案的结果应导致在集合理论和一般拓扑中发展新的重要思想，我们希望能在数学的其他领域找到应用。该提案中提出的问题引起了该领域的专家，不仅在加拿大而且在国际上也是如此。除了扩大有关拓扑空间知识的领域外，该研究建议还将继续促进加拿大作为纯数学研究中心。