E-6 - Baire Spaces

Abstract

A topological space X is a Baire space if every intersection of countably many dense open sets in X is dense in X. A meager set (or set of first category) can be covered by countably many closed sets with empty interior. In many important cases, the objects under consideration form a set in a suitable Baire space. If so, these objects can be viewed as “typical”, or “generic”, even if their individual construction may be fairly complicated. This chapter presents a sample of some significant instances of this phenomenon. Namioka's Theorem states that if : X ×K → ℝ is a separately continuous real valued function defined on the product of the Čech-complete space X and the compact space K, that is, for every x the function ƒ(x, ) is continuous, then there is a comeager set A ⊂X such that ƒ is jointly continuous at each point of A ×K. The complement of a Souslin set in the square—all of whose vertical sections are non-meager—contains the graph of a Borel function. The category transforms, introduced by R. Vaught, play an important role in the study of Borel theory of group actions.