A Partition Theorem of ωωα

Abstract

We consider finite partitions of the closure $\overline{\mathcal{F}}$ of an ${\omega }^{\alpha }$-uniform barrier $\mathcal{F}$. For each partition, we get a homogeneous set having both the same combinatorial and topological structure as $\overline{\mathcal{F}}$, seen as a subspace of the Cantor space $2^{\mathbb{N}}$.