## Abstract

We study the Baire type properties in the classes B_{α}(X,Y) of Baire-α functions and B_{α} ^{st}(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 B_{1}(X)=R^{X} iff X is a Q-space. If X is a Tychonoff space of countable pseudocharacter, then: (i) for every ordinal α≥2, the spaces B_{α}(X) and B_{α} ^{st}(X) are Choquet and hence Baire, and (ii) B_{1}(X) is a Choquet space iff X is a λ-space. Also we prove that for a Tychonoff space X the space B_{1}(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) B_{1}(X) is a Baire space, (ii) B_{1}(X) is a Choquet space, and (iii) every compact subset of X is scattered. We show that the Baireness of the function space B_{1}(X) is essentially a property of separable subspaces of X: if Y is a Polish space and X is a Y-Hausdorff space, then B_{1}(X,Y) is Baire iff every countable subset of X is contained in a subspace Z⊆X such that the function space B_{1}(Z,Y) is Baire and the restriction operator B_{1}(X,Y)→B_{1}(Z,Y), f↦f|_{Z}, is surjective.

Original language | English |
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Article number | 107078 |

Journal | Topology and its Applications |

Volume | 272 |

DOIs | |

State | Published - 1 Mar 2020 |

## Keywords

- Airy space
- Almost K-analytic
- B (X,Y)
- B(X,Y)
- Baire space
- Choquet space
- Q-space
- Scattered
- The first Baire class
- Čech-complete
- λ-Space