Elementary Facts about Baire and Baire-Type Spaces

Abstract

This chapter contains classical results about Baire-type conditions (Baire-like, b-Baire-like, CS-barrelled, s-barrelled) on tvs. We include applications to closed graph theorems and C(X) spaces. We also provide the first proof in book form of a remarkable result of Saxon (extending earlier results of Arias de Reyna and Valdivia), that states that, under Martin’s axiom, every lcs containing a dense hyperplane contains a dense non-Baire hyperplane. This part also contains analytic characterizations of certain completely regular Hausdorff spaces X. For example, we show that X is pseudocompact, is Warner bounded, or C _c (X) is a (df)-space if and only if for each sequence (μ _n ) _n in the dual C _c (X)′ there exists a sequence (t _n ) _n ⊂(0,1] such that (t _n μ _n ) _n is weakly bounded, strongly bounded, or equicontinuous, respectively. These characterizations help us produce a (df)-space C _c (X) that is not a (DF)-space, solving a basic and long-standing open question.