Compact spaces of Szlenk index ω

The Szlenk index has found many applications in the isomorphic theory of Banach spaces. Its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure. There is no obstacle to consider the notion of Szlenk index in more general settings. In this paper we study the compact spaces of Szlenk index ω at most with respect to an associated metric. We include new applications to Banach spaces of the these methods, where the estimations of the growth speed of the finite Szlenk indices play a fundamental role.