Some problems from set-theoretic topology - normality, D-spaces and homogeneity

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Since its establishment in the middle of the last century, the field of set-theoretic topology has stood on the boundary between to foundational areas of pure mathematics: General Topology and Set Theory. The language and theory of sets is, indeed, the setting in which all of mathematics is axiomatized and the language in which fundamental questions around logic, truth and consistency are formalized. Godel's incompleteness theorem tells us that any axiomatic foundation for mathematics will include statements independent of the axioms (i.e., neither provable nor refutable). And indeed, the first area of mathematics where important and natural open problems turned out to be independent was in Topology an area of mathematics with a geometric flavour that is fundamental basis for areas of mathematics such as Analysis. This phenomenon of independence results has turned out to be quite endemic, even quite recently arising in Physics (with the proof of the independence of the general spectral gap problem). ***Set-theoretic topology is still an area where the interplay between both areas give rise to advances and the development of techniques that have impact and applications to many other areas of mathematics. For example, many important combinatorial tools (e.g., forcing techniques, Ramsey theory, combinatorial principles extracted from Godel's constructible universe, etc) that were initially developed to solve problems arising from Topology then found applications in areas as diverse as Functional Analysis, Group Theory and combinatorics. ******In 2007, M. Hrusak and J. Moore compiled a list “Twenty problems in set-theoretic topology” the most important and long-standing open problems in the field. This list, which formed the introduction to the monograph, Open Problems In Topology, was in the spirit of Hilbert's famous 100 problems formulated at the ICM in 1900 and the more recent Clay Institute Millennial problems, which were meant to encourage the mathematical community to focus and collaborate on the most important and impactful open problems. My proposed program of study is organized around three connected problems highlighted in the Hrusak-Moore list. Mary Ellen Rudin's problem whether there is a ``small'' Dowker space and the problems of van Douwen: Are Lindelof regular spaces D-spaces? And is the continuum the a bound on the cellularity of compact homogeneous spaces? These problems have been open for many decades and underscore the fact that combinatorics lie at the heart of many fundamental questions about the structure of topological spaces. Moreover, the many surprising connections between these problems and other topological and set theoretic questions explain their importance in the field and their prominence on the Hrusak-Moore list. The solution of any of these three problems will involve the development of new techniques and ideas which are certain to find applications in the field and inevitably to other areas of mathematics.