Problems concerning convergence and separation properties in topological spaces

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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.