I don't understand: why is this so? I've just seen the proof that a $T_0$ topological group is $T_1$, but don't know how to show that it's $T_{3.5}$.
BTW, the fact that $x\overline{V}=\overline{xV}$, one inclusion is obvious $\overline{xV} \subset x\overline{V}$, because $xV \subset x\overline{V}$, and $\overline{xV}$ is the least closed set that contains $xV$.
Now the reverse inclusion if I take $y \in x\overline{V}$, then $y=xv$ for some $v \in \overline{V}$, which means there's an open set $O$, s.t $v \in O\cap V \neq \emptyset$, now we have $xv \in x(O\cap V)= (xO \cap xV)$, so $y \in \overline{xV}$, is this about right?
I'm more in need of help with the $T_{3.5}$ property, if you have any reference for this proof, works for me as well.
Thanks.