Finite Combinations of Baire Numbers

Abstract

Let $κ$ be a regular cardinal. Consider the Baire numbers of the spaces $(2^θ)_κ$ (functions from $θ$ to 2 and the less than $κ$ topology) for various $θ\geq κ$. 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 $κ> ω$, then l could be any given finite number. We do not know whether the same is true for $κ= ω$.