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: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508502
• 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.

该建议致力于研究纯数学，拓扑学和集合论两个领域的交叉点。 拓扑思想是数学领域的核心，其中距离的概念发挥了作用，并关注那些不受保持接近关系的变化或变形影响的抽象几何对象的属性。这是一个美丽的数学领域，具有强大的力量来简化和澄清看似不同领域的想法。例如，微积分的基本概念是拓扑思想的简单结果。另一方面，集合论构成了整个数学的基础。集合论的一个典型的基础问题是：“我们使用的和数学中需要的基本假设（公理）是什么？”哥德尔不完备性定理的一个推论是，对于任何可接受的数学公理系统，都有一些数学陈述既不能被证明，也不能从系统中被反驳。知道一个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（知道一个猜想是独立的意味着我们可以停止寻找它的证明），而且它还建立了与集合论领域的明确联系。在明显不同的数学领域之间建立联系总是产生有趣和重要的数学。这个研究建议的重点是关于拓扑空间的基本结构的问题，其中深集理论的连接已经建立或我们认为它们存在。 这一建议的结果应导致发展的新的和重要的想法，在集理论和一般拓扑，我们希望将找到应用到其他领域的数学。本提案中提出的问题不仅对加拿大而且对国际上的这一领域的专家都很感兴趣。除了扩大有关拓扑空间的知识前沿，这项研究计划将继续促进加拿大作为纯数学研究中心。