Density and Baire category in recursive topology

摘要

We develop the concepts of recursively nowhere dense sets and sets that are recursively of first category and study closed sets of points in light of Baire's Category Theorem. Our theorems are primarily concerned with exdomains (i. e. complements of domains) of recursive quantum functions and hence with avoidable points (i. e. points that can be forced into exdomains of recursive quantum functions). An avoidance function is a recursive function which can be used to expel avoidable points from domains of recursive quantum functions. We define an avoidable set of points to be an arbitrary subset of the avoidable points of a single avoidance function, and we study an effective union of such sets which we call a piecemeal avoidable set. We note that each of the set of recursive points and the set of avoidable points is of first category but not recursively of first category. We show an exdomain exists which is recursively nowhere dense, as well as one that is nowhere dense but not recursively nowhere dense. After establishing that every exdomain is recursively of first category, we prove that given any fixed exdomain, there is another exdomain, which while dense in the underlying space, is disjoint from the fixed exdomain. Finally, we show how to build a recursive sequence of recursive quantum functions that have mutually disjoint, dense exdomains.