I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic?


The reason I'm asking this is because I was wondering whether or not there were any disconnected real topological vector spaces.



Are all topological (finite-dim) real vector spaces homeomorphic to a coordinate space?
Any (Hausdorff) topological real vector space of dimension $n<\infty$ is homeomorphic to $\mathbf{R}^n$ with the standard topology, see e.g. Rudin, Functional analysis, theorem 1.21.


Here are some comments:


1. For some reason it is stated there for complex vector spaces, but, as remarked after the theorem, the proof works for real vector spaces as well.
2. Instead of the Hausdorff axiom Rudin uses the (weaker) $T\_1$ axiom in the definition of a topological vector space.


