  Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces
$(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$
topology) for various $\theta \geq \kappa$. Let l be the number of such
different Baire numbers. Models of set theory with l=1 or l=2 are known and it
is also known that l is finite. We show here that if $\kappa > \omega$, then l
could be any given finite number. We do not know whether the same is true for
$\kappa = \omega$.
