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Let $V$ be a finite-dimensional real vector space. Eric Wofsey's answer to this post claims

The usual topology on $V$ is the unique topology that makes $V$ a topological vector space. That is, it is the unique $T_0$ topology that makes addition $V\times V\to V$ and scalar multiplication $\mathbb R\times V\to V$ continuous.

He cites this post for details, but as far as I can tell, the answer there only proves the analogous statement for Hausdorff ($T_2$) topologies. I know that $T_1$ and $T_2$ are equivalent on a TVS, so the linked answer also proves the above claim with $T_0\to T_1$. But I can't find a proof for the $T_0$ case. Please prove the above claim.

WillG
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    We just need to know that a $T_0$ topological group is Hausdorff, for which see: https://math.stackexchange.com/questions/1593365/a-t-0-topological-vector-space-is-hausdorff – Qiaochu Yuan Sep 12 '22 at 00:35

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