Abstract:This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space of irrationals, or certain of its subspaces. In particular, given $f\in {}^\omega (\omega \setminus \{0\})$, we consider compact sets of the form $\prod_{i\in \omega }B_i$, where $|B_i|= f(i)$ for all, or for infinitely many, $i$. We also consider ``$n$-splitting'' compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega $, $|\{g(i):g\in K, g\restriction i=f\restriction i\}|= n$.
Keywords: irrationals, $f$-cone, weak $f$-cone, $n$-splitting compact set
AMS Subject Classification: 03E17, 03E35, 54A35