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How to prove the following :

Given $(X, \tau)$ a topological vector space on $\mathbb{R}$, If {0} is closed then $\tau$ is separated.

Thank you.

user2015
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    This holds for any topological group, and relies on the fact that if $x \neq y$, then there exists two open sets separating $0$ and $x-y$. – Crostul Dec 17 '15 at 12:34

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