The Baire theory of category, which classifies sets into two distinct categories, is an important topic in the study of metric spaces. Many results in topology arise from category theory; in particular, the Baire categories are related to a topological property. Because the Baire Category Theorem involves nowhere dense sets in a complete metric space, this paper first develops the concepts of n...

We obtain game–theoretic characterizations for meagerness and rareness of filters on ω. One of the classical methods for obtaining a set of real numbers which does not have the property of Baire, is to interpret appropriate filters on N = {1,2,3, . . . } as subsets of [0, 1]. Filters which result in a set having the property of Baire have nice combinatorial characterizations, due to Talagrand [...

In a recent paper E. J. McShane [3]2 has given a theorem which is the common core of a variety of results about Baire sets, Baire functions, and convex sets in topological spaces including groups and linear spaces. In general terms his theorem states that if J is a family of open maps defined in one topological space Xi into another, X2, the total image JiS) of a second category Baire set S in ...

There is a well-known global equivalence between Σ2 sets having the Universal Baire property, two-step Σ3 generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of Σ2 sets being (2 )-cc Universally Baire, which is below 0. In a model obtained, there is a Σ2 set which is weakly ω2-Universally Baire but not ω2-Uni...

It is proved that every function of finite Baire index on a separable metric space K is a D-function, i.e., a difference of bounded semi-continuous functions on K. In fact it is a strong D-function, meaning it can be approximated arbitrarily closely in D-norm, by simple D-functions. It is shown that if the n derived set of K is non-empty for all finite n, there exist D-functions on K which are ...