Mathematical Sciences: Investigations on Normal and Paracompact Spaces

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9623391 Balogh Although normal and paracompact spaces are classical and frequently used classes of topological spaces, and have been extensively investigated (indeed, major breakthroughs such as those on normal Moore spaces and Dowker spaces took place exactly in this area), they are also among those classes in which a number of old and venerable problems remain open. The aim of this project is to make progress in settling some of these problems, mainly by creating new constructions of topological spaces. The principal technique to be used has its origin in a paper of M.E.Rudin, where an example of a normal but not collectionwise normal simplicial complex is constructed. In that paper Rudin uses countable substructures of a structure of size 2^c on a set of cardinality c (= the cardinality of the real line) to code normality. Refinements of and improvements on this technique will be used to attack seven problems from the literature as well as to obtain related results. A typical such problem is whether and which normal, screenable (= every open cover has a sigma-disjoint open refinement) spaces are paracompact. The following two important properties of the usual Euclidean space are often used in mathematics. (1) Every continuous function on a closed set can be continuously extended over the entire space (normality). (2) Continuous local structures can be continuously amalgamated into global ones (paracompactness). Research during and before the early fifties showed that many of the more general types of spaces used in mathematical applications also have one or both of these properties; those that do are called, respectively, normal and paracompact spaces. Major breakthroughs in the seventies and eighties uncovered much of the structure of normal and paracompact spaces. However, basic research questions remain unanswered. Such questions are typically concrete, explicit conjectures in the literature that have to do with problems such as these: (a ) How does one recognize paracompact spaces in the class of normal spaces? (b) Does the product of two or more normal spaces of a certain kind also have this useful extension property? During the course of this project, refinements of a technique originated by M.E.Rudin will be used to build general topological spaces, with the goal of settling some of the conjectures mentioned above. ***