Some Questions in Set-theoretic Topology

Auburn University, Auburn AL

Investigators

Abstract

Proposal: DMS-0072269 PI: Gary Gruenhage Abstract: The principle investigator proposes to focus on three well-known open problems in set-theoretic topology: M. Husek's 1977 question of whether compact spaces having a small diagonal are metrizable, J. Steprans' fifteen-year old problem concerning the existence of uncountable non-discrete spaces which are homeomorphic to each of their uncountable subspaces, and J. Ceder's forty year old problem of whether stratifiable spaces have a sigma-closure preserving base. All three questions involve fundamental topological properties and constructions; they lie in the mainstream of set-theoretic topology and have been worked on by many researchers. The principle investigator has significant partial results on all of them which answer related questions and give hope for further progress. In particular, he has settled the first question for countably compact spaces, affirmatively answered a countable version of the second question, and with respect to the third question has obtained several important results including a positive answer to a closely related question of Ceder. Set theory and general topology are fundamental mathematical disciplines with common turn-of-the-century roots. General topology provides a framework for the study of shapes, from ordinary structures in standard three-dimensional space to much more abstract shapes, and the continuous deformations of these shapes. Dramatic advances in set theory and logic during the past forty years have led to the realization that many long-standing questions in the general topology of abstract spaces rest on complicated set-theoreti combinatorics and often cannot be solved assuming only the standard axioms of set theory. Thus began a fruitful interaction between the set theory and general topology which continues today. The principle investigator's proposed problems lie within the scope of this interesting interaction. Proposed methods of solutions to the problems involve an interweaving of topological methods with sophisticated set theoretic tools. Solutions to these problems would deepen our understanding of fundamental topological properties and relationships, and would likely require new set-theoretical and topological techniques applicable to a variety of other problems.