Baire category properties of some Baire type function spaces

Abstract

We study the Baire type properties in the classes Bα(X,Y) of Baire-α functions and Bαst(X,Y) of stable Baire-α functions from a topological space X to a topological space Y, where α≥1 is a countable ordinal. Among others we prove the following results. If X is a normal space, then B1(X)=RX iff X is a Q-space. If X is a Tychonoff space of countable pseudocharacter, then: (i) for every ordinal α≥2, the spaces Bα(X) and Bαst(X) are Choquet and hence Baire, and (ii) B1(X) is a Choquet space iff X is a λ-space. Also we prove that for a Tychonoff space X the space B1(X) is meager if X is airy. We introduce a new class of almost K-analytic spaces which properly contains Čech-complete spaces and K-analytic spaces and show that for an almost K-analytic space X the following assertions are equivalent: (i) B1(X) is a Baire space, (ii) B1(X) is a Choquet space, and (iii) every compact subset of X is scattered. We show that the Baireness of the function space B1(X) is essentially a property of separable subspaces of X: if Y is a Polish space and X is a Y-Hausdorff space, then B1(X,Y) is Baire iff every countable subset of X is contained in a subspace Z⊆X such that the function space B1(Z,Y) is Baire and the restriction operator B1(X,Y)→B1(Z,Y), f↦f|Z, is surjective.