Problems concerning convergence and separation properties in topological spaces

拓扑空间中的收敛性和分离性问题

• 批准号: 238944-2006
• 负责人: Szeptycki, Paul
• 金额: $ 1.09万
• 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: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=394138
• 关键词: Problems; concerning; convergence; separation; properties

Set theory and topology are two research areas that are sometimes described as a part of the "foundations" of mathematics. A typical foundational question one might ask is "what assumptions (axioms) are needed in order to develop modern mathematics?"  The axioms of Zermelo-Frankel set theory with the Axiom of Choice are now almost universally accepted as the axiom system within which modern mathematics is developed. A consequence of Godel's Incompleteness Theorem is that ZFC, and for that matter any other acceptable axiom system for mathematics, must be incomplete: there are mathematical statements that are independent of ZFC, i.e., they can neither be proven nor refuted from the axioms of ZFC. There are now many important mathematical conjectures from across many mathematical disciplines that have been shown to be independent of ZFC. And the area of topology has more than its fair share. Knowing that a particular mathematical conjecture is independent is not only of obvious intrinsic interest (knowing it is indendent means that we can stop looking for a proof of the conjecture or of the negation of the conjecture) but establishes a clear connection between one mathematical discipline to the area of set theory. Establishing connections between apparantly disparate mathematical fields has always produced interesting and important mathematics. Research related to independence results in topology has played an important role in the development of research in both areas of set theory and topology. This research proposal is focussed on some particular problems from general topology where deep set theoretic connections have either already been established, or we believe exist. It is expected that work on this proposal should lead to the development of new and important ideas in both set theory and in general topology.

集合理论和拓扑是两个研究领域，有时被描述为数学“基础”的一部分。人们可能会问的一个典型的基础问题是“为了开发现代数学，需要哪些假设（公理）？” Zermelo-frankel Set理论具有所选公理的公理现在几乎被普遍接受为开发现代数学的公理系统。 Godel不完整定理的结果是，ZFC，而对于任何其他可接受的数学公理系统，都必须不完整：有些数学陈述独立于ZFC，即可以被证明，也可以从ZFC的公理中得到证明，也可以被驳斥。现在有许多来自许多数学学科的数学猜想，这些猜想已被证明与ZFC无关。拓扑领域的份额不仅仅是其公平份额。知道特定的数学猜想是独立的，不仅具有明显的内在兴趣（知道这是独立的，这意味着我们可以停止寻找协议的证明或协议的谈判证明），而且在与集合理论领域的一项数学学科之间建立了明确的联系。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。与独立性结果相关的研究在整体理论和拓扑领域的研究发展中发挥了重要作用。这项研究建议集中在一般拓扑的一些特定问题上，这些问题已经建立了深度的理论和一般拓扑，或者我们认为存在。可以预期，该提案的工作应导致在设定理论和一般拓扑中的新重要思想的发展。