Some Questions in Set-theoretic Topology

摘要：主要研究者提出集中研究集合论拓扑中三个著名的开放问题：M. Husek在1977年提出的具有小对角线的紧空间是否可度量的问题，J. Steprans在15年前提出的关于不可数-离散空间与它们的不可数子空间同胚的存在性的问题，以及J. Ceder在40年前提出的可分层空间是否有sigma闭包保持基的问题。这三个问题都涉及基本的拓扑性质和结构；它们属于集合论拓扑学的主流，已被许多研究者所研究。主要研究者都有显著的部分结果，这些结果回答了相关的问题，并给了进一步进展的希望。特别是，他解决了可数紧空间的第一个问题，肯定地回答了第二个问题的可数版本，关于第三个问题，他得到了几个重要的结果，包括对一个密切相关的Ceder问题的肯定答案。集合论和一般拓扑学是基本的数学学科，具有共同的世纪之交根源。一般拓扑学为研究形状提供了框架，从标准三维空间中的普通结构到更抽象的形状，以及这些形状的连续变形。在过去的四十年中，集合论和逻辑学的巨大进步使人们认识到，在抽象空间的一般拓扑中，许多长期存在的问题依赖于复杂的集合论组合，并且往往不能仅假设集合论的标准公理来解决。于是，集合论和一般拓扑学之间开始了卓有成效的相互作用，这种相互作用一直持续到今天。主要研究者提出的问题就在这个有趣的相互作用的范围内。所提出的解决问题的方法包括拓扑方法与复杂的集合理论工具的交织。这些问题的解决方案将加深我们对基本拓扑性质和关系的理解，并且可能需要适用于各种其他问题的新的集合理论和拓扑技术。

Proposal: DMS-0072269PI: Gary GruenhageAbstract:The principle investigator proposes to focus on three well-known openproblems in set-theoretic topology: M. Husek's 1977 question of whethercompact spaces having a small diagonal are metrizable, J. Steprans'fifteen-year old problem concerning the existence of uncountablenon-discrete spaces which are homeomorphic to each of their uncountablesubspaces, and J. Ceder's forty year old problem of whether stratifiablespaces have a sigma-closure preserving base. All three questions involvefundamental topological properties and constructions; they lie in themainstream of set-theoretic topology and have been worked on by manyresearchers. The principle investigator has significant partial resultson all of them which answer related questions and give hope for furtherprogress. In particular, he has settled the first question for countablycompact spaces, affirmatively answered a countable version of the secondquestion, and with respect to the third question has obtained severalimportant results including a positive answer to a closely relatedquestion of Ceder. Set theory and general topology are fundamental mathematical disciplineswith common turn-of-the-century roots. General topology provides aframework for the study of shapes, from ordinary structures in standardthree-dimensional space to much more abstract shapes, and the continuousdeformations of these shapes. Dramatic advances in set theory and logicduring the past forty years have led to the realization that manylong-standing questions in the general topology of abstract spaces rest oncomplicated set-theoreti combinatorics and often cannot be solvedassuming only the standard axioms of set theory. Thus began a fruitfulinteraction between the set theory and general topology which continuestoday. The principle investigator's proposed problems lie within thescope of this interesting interaction. Proposed methods of solutions tothe problems involve an interweaving of topological methods withsophisticated set theoretic tools. Solutions to these problems woulddeepen our understanding of fundamental topological properties andrelationships, and would likely require new set-theoretical andtopological techniques applicable to a variety of other problems.