Motivation
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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.




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So here are my two well-posed questions:



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> a) Suppose *K* and *F* are topological fields. If *K* and *F* are homeomorphic, then is it necessarily true that *char K* = *char F*?
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> b) Suppose *K* and *F* are topological fields. Further suppose that $V$ is a topological vector space over *K* and $W$ is a topological vector space over *F*. If $V$ and $W$ are homeomorphic, then is it necessarily true that *char K* = *char F*?
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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.



Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
I think all non-archimedean locally compact fields are homeomorphic: Their rings of integers are compact, metric and totally disconnected and hence are all homeomorphic (to the Cantor set). The same is true for the units in those rings. The field is then topologically the disjoint union of the ring and a countable number of copies of the units.


