If $A$ is a topological vector space, then I know that being $T_1$ is equivalent to beeing $T_2$.
Now I was wondering if $A$ is a $T_0$-space, is it then automatically $T_1$ (and therefore $T_2$).
I can't find an easy counterexample making me think that it might be true.
2026-03-28 03:02:23.1774666943
Is a topological vector space that is $T_0$ already $T_2$?
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This is true more generally for arbitrary topological groups. Indeed, suppose $A$ is $T_0$. Given two distinct $x,y$ there is a neighbourhood $U$ of one not containing the other, say $x\in U,y\not\in U$. Then $y+x-U$ is a neighbourhood of $y$ not containing $x$ (my comment above has a typo). Therefore $A$ is $T_1$ which, as you already know, implies $T_2$.