On topological classification of non-archimedean Fréchet spaces

summary:We prove that any infinite-dimensional non-archimedean Fréchet space $E$ is homeomorphic to $D^{\mathbb{N}}$ where $D$ is a discrete space with $\mathop {\mathrm card}(D)=\mathop {\mathrm dens}(E)$. It follows that infinite-dimensional non-archimedean Fréchet spaces $E$ and $F$ are homeomorphic if and only if $\mathop {\mathrm dens}(E)= \mathop {\mathrm dens}(F)$. 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 $\mathbb{K}^{\mathbb{N}}$