Compact subsets of $\Ps(\N)$ with applications to the embedding of symmetric sequence spaces into $C(\alpha )$

Abstract

Let $\Ps(\N)$ be the set of all finite subsets of $\mathbb{N}$, endowed with the product topology. A description of the compact subsets of $\Ps(\N)$ is given. Two applications of this result to Banach space theory are shown : (1) a characterization of the symmetric sequence spaces which embed into $C(\om^\om)$, and (2) a characterization, in terms of the Orlicz function $M$, of the Orlicz sequence spaces $h_M$ which embed into $C(K)$ for some countable compact Hausdorff space $K$.