Problems concerning convergence and separation properties in topological spaces

集合论和拓扑学是两个研究领域，有时被描述为数学“基础”的一部分。一个典型的基本问题可能会问：“为了发展现代数学，需要什么样的假设（公理）？“Zermelo-Frankel集合论的公理与选择公理现在几乎被普遍接受为现代数学发展的公理系统。哥德尔不完备性定理的一个结果是ZFC，以及任何其他可接受的数学公理系统，必须是不完备的：存在独立于ZFC的数学陈述，即，它们既不能从ZFC的公理中被证明也不能被反驳。现在有许多来自许多数学学科的重要数学命题已经被证明是独立于ZFC的。拓扑学的领域有超过其公平份额。知道一个特定的数学猜想是独立的，不仅具有明显的内在利益（知道它是独立的，意味着我们可以停止寻找猜想的证明或猜想的否定），而且在一个数学学科与集合论领域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总是产生有趣和重要的数学。拓扑学中与独立性相关的研究结果在集合论和拓扑学的研究发展中发挥了重要作用。这项研究建议的重点是一些特殊的问题，从一般拓扑学的深集理论的连接已经建立，或者我们相信存在。预计这项工作的建议应导致发展的新的和重要的想法都集理论和一般拓扑结构。

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.