Completeness and reflexivity type properties of $B_1(X)$

TL;DR

This work characterizes the topological and functional-analytic properties of the Baire-one function space $B_1(X)$ on a Tychonoff space $X$ by linking completeness, reflexivity, and Montel-type conditions to structural properties of $X$, notably forming CZ-spaces and $Q_f$-spaces. It proves that, for spaces with countable pseudocharacter, a spectrum of properties (Montel, reflexive, complete, and $B_1(X)=\mathbb{R}^X$) are equivalent to $X$ being a $Q_f$-space, while sequential and local completeness of $B_1(X)$ coincide with $X$ being CZ-space. It also treats compact and separable metrizable cases: $B_1(K)$ is locally complete iff $K$ is scattered, and for separable metrizable $X$, $B_1(X)$'s regularity reflects whether $X$ is a $Q$-set, $\sigma$-set, $\lambda$-set, or $\kappa$-set, with connections to small cardinals. These results illuminate the deep interplay between descriptive set-theoretic properties of $X$ and the topological behavior of $B_1(X)$. The paper also provides numerous examples and discusses conditions under standard set-theoretic axioms.

Abstract

For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2: BG-Baire-lcs
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof