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 \({\mathbb{K}}^{\mathbb{N}} \).