## Motivation

A *topological vector space* is a vector space over a (topological) field, *K*, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector spaces. Since $\mathbb{C} \approx \mathbb{R}^2$ as topological spaces (with their standard topologies), it's clear that the actual field cannot be detected by the topology alone. Also, cardinality considerations show that any topology on a finite field cannot be homeomorphic to a vector space over a field of characteristic $0$. But that still says nothing about whether there are some topologies, such that, for instance $\mathbb{Z}_p(t) \approx \mathbb{Q}$. I can't find any topological invariants that will distinguish just the fields (as vectors spaces of dimension 1). I suspect that if this problem is solvable, it will be just be topological group considerations, but I can't figure out how to do it.

So here are my two well-posed questions:

a) Suppose

KandFare topological fields. IfKandFare homeomorphic, then is it necessarily true thatchar K=char F?b) Suppose

KandFare topological fields. Further suppose that $V$ is a topological vector space overKand $W$ is a topological vector space overF. If $V$ and $W$ are homeomorphic, then is it necessarily true thatchar K=char F?

Obviously, an answer to b) implies an answer for a). I'm not sure how much of my thinking I should put here. But I do know that if *char K* = *p*, then we get a homeomorphism of $V$ onto itself by translation for each $v \in V$, which each have finite order *p*. These are mapped to self-homeomorphisms of $W$ which also have order *p*, but a priori, they don't "see" any of the algebraic structure of $W$, so I'm not sure this is actually an obstruction. Any thoughts (or references) to proofs or counterexamples would be greatly appreciated.