Lindelöf-type properties of spaces of Baire-one functions

In this paper, we investigate Lindelöf-type properties of spaces of Baire-one functions equipped with the topology of pointwise convergence. In particular, we investigate the properties of normality, extent and realcompactness for the space $B_1(X)$ of Baire-one real-valued functions. We also study when $B_1(X)$ is realcompact: we show that this is the case for every hereditarily Baire (in particular, every Polish) separable metrizable space $X$ and for every Luzin set $X$. The paper provides answers to two Teng’s questions. The paper raises open questions.