On Topological Classification of Non-Archimedean Fréchet Spaces

Abstract

We prove that any infinite-dimensional non-archimedean Fréchet space E is homeomorphic to $D^{\mathbb{N}}$ where D is a discrete space with card(D) = dens(E). It follows that infinite-dimensional non-archimedean Fréchet spaces E and F are homeomorphic if and only if dens(E) = dens(F). In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field ${\mathbb{K}}$ is homeomorphic to the non-archimedean Fréchet space ${\mathbb{K}}^{\mathbb{N}}$.