Problems related to D-spaces.

本提案致力于研究两个纯数学领域的交叉，拓扑和集合论。拓扑学思想是距离概念发挥作用的数学领域的核心，它关注的是那些不受变化或变形影响的抽象几何对象的属性，这些属性保持了接近关系。这是一个美丽的数学领域，它具有强大的力量，可以简化和澄清看似不同的领域的想法。例如，微积分中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。集合论中一个典型的基础问题是，“我们在数学中使用和需要的基本假设（公理）是什么？”哥德尔不完备定理的一个结论是，对于任何可接受的数学公理系统，都存在既不能从该系统证明也不能从该系统驳斥的数学命题。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利益（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），而且它还与集合论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总是会产生有趣而重要的数学。本研究计划的重点是关于拓扑空间的基本结构问题，其中深度集论连接要么已经建立，要么我们相信它们存在。这一建议的结果将导致新的和重要的思想发展，在集合论和一般拓扑，我们希望将找到应用到数学的其他领域。这一建议中提出的问题不仅对加拿大而且对国际上这一领域的专家都很有兴趣。除了拓展拓扑空间的前沿知识外，这项研究计划将继续推动加拿大成为纯数学研究中心。

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.