Is the standard structure of a topological vector space on reals unique? The standard stucture of a topological vector space on reals is this given by the metric d(x,y)=|x-y| on the vector space $\mathbb{R},$ with the field of scalars $\mathbb R$ with standard topology.
I would like to know if we can change the topology on the reals (considered as a vector space, not a field) and still obtain the structure of a topological vector space. To be clear, the only thing I want to change is the topology of the topological vector space. I don't want to change any operations, or the field of scalars, or the topology on the field of scalars.
 A: If you don't require the topology to be $T_0$, you can simply take the trivial topology on $\mathbb{R}$.
If you do require it to be $T_0$, then it must be $T_2$ as we know from the general theory. The nice argument of Davide Giraudo (see comments to the original post) shows in this case that any such topology $\mathcal{T}$ on $\mathbb{R}$ must contain every usual open set, that is $\mathcal{T_\mathrm{usual}}\subseteq\mathcal{T}$. Now you can show, if I haven't made any mistakes, that the usual topology is the only one that works, by reductio ad absurdum:
Suppose there is some $U\in\mathcal{T}$ that is not open in the usual topology. Let $F = U^C$ be its complement and let $M:\mathbb{R}_\mathrm{usual}\times\mathbb{R}\to\mathbb{R}$ be the scalar multiplication, where $\mathbb{R}$ is equiped with the topology $\mathcal{T}$ and $\mathbb{R}_\mathrm{usual}$ has the usual topology. Since $F$ is closed, $M^{-1}(F) =\lbrace(x,y)|\hbox{ }xy\in F\rbrace$ is closed and therefore its intersection with $\mathbb{R}\times\lbrace1\rbrace$ is also closed in the product topology. But $M^{-1}(F)\cap\mathbb{R}\times\lbrace1\rbrace = \lbrace(x,1)|\hbox{ }x\in F\rbrace = F\times\lbrace 1\rbrace$, so $F$ would have to be closed in the usual topology. This is not the case, so this is a contradiction.
Conclusion: there is only one such Hausdorff topology on $\mathbb{R}$, the usual one.
