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 ...

We study the uniform computational content of different versions of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a complete (i.e., “large”) metric space cannot be decomposed into countably many nowhere dense (i.e., “small”) pieces. The Baire Category Theorem is an illuminating example of a theorem that can...

In this notes we talk about the Baire category theorem and its consequences: the BanachSteinhaus theorem, the open mapping theorem and the closed graph theorem.

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bangbang property is proved for a new class of “concave” multifunctions, characterized by the existence of suitable linear selections. The proofs rely ...

Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.