separable in topology of $E$

Question

let $S$ be a semigroup and $J$ be a finite subset of $S$. let $I$ denote the left ideal generated by $J$ and $J$ action on weakly closed convex subset $C$ of locally convex space $E$. if $C(x)(J) $denote the smallest convex and weakly closed set which $I$ invariant , where $x$ is arbitrary element of $C$, why $C(x)(J) $ are separable in topology of $E$?