On the collection of Baire class one functions on the irrationals

Abstract:

The Baire class one fan over the irrationals $\mathbb {N} ^{\mathbb {N}}$ is the space $S(\mathbb {N} ^{\mathbb {N}})$ $= ( \mathbb {N} ^{\mathbb {N}} \times \mathbb {N})\cup \{ \infty \}$, where basic neighbourhoods of $\infty$ are epigraphs of the first Baire class functions $f: \mathbb {N}^{\mathbb {N}}\to \mathbb {N}$, augmented by $\infty$, and the remaining points are isolated; $S(\mathbb {N}) = ( \mathbb {N} \times \mathbb {N} )\cup \{\infty \}$ is the standard countable sequential fan. We prove that $S(\mathbb {N}^{\mathbb {N}}) \times S(\mathbb {N})$ has countable tightness: in this product, whenever $p \in \overline {A}$, then $p\in \overline {B}$ for some countable $B\subset A$.