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

TL;DR

This paper characterizes when the space of Baire-one functions on a Tychonoff space is complete, reflexive, or Montel, linking these properties to the topological nature of the underlying space.

Contribution

It establishes equivalences between various topological properties of $B_1(X)$ and the structure of the space $X$, including conditions for completeness and reflexivity.

Findings

$B_1(X)$ is Montel iff it is reflexive iff it is complete iff $X$ is a $Q_f$-space.

Sequential completeness of $B_1(X)$ is equivalent to local completeness and $X$ being a $CZ$-space.

For compact $K$, $B_1(K)$ is locally complete iff $K$ is scattered.

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.