Some Questions in Set-theoretic Topology

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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.