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。人们可能会问的一个典型的基础问题是“为了发展现代数学，需要什么假设（公理）？”Zermelo-Frankel集合论的公理和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。哥德尔不完备定理的一个结论是，ZFC，以及任何其他可接受的数学公理系统，都必须是不完备的：存在独立于ZFC的数学命题，即它们既不能被证明，也不能被ZFC的公理反驳。现在有许多重要的数学猜想来自许多数学学科，它们被证明是独立于ZFC的。拓扑学的面积比它的公平份额大得多。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能产生有趣而重要的数学。拓扑学中独立性结果的研究对集合论和拓扑学的发展都起着重要的作用。本研究计划的重点是一般拓扑学中的一些特殊问题，其中深度集论连接要么已经建立，要么我们认为存在。人们期望在这一建议上的工作将导致在集合论和一般拓扑中新的和重要的思想的发展。

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.