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We determine the closure of a cyclic subgroup $H$ of a free group for thepro-{\bf V} topology when {\bf V} is an extension-closed pseudovariety offinite groups. We show that $H$ is always closed for the pro-nilpotent topologyand compute its closure for the pro-$\mathbf{G}_p$ and pro-$\mathbf{V}_p$topologies, where $\mathbf{G}_p$ and $\mathbf{V}_p$ denote respectively thepseudovariety of finite $p$-groups and the pseudovariety of finite groupshaving a normal Sylow $p$-subgroup with quotient an abelian group of exponentdividing $p-1$. More generally, given any nonempty set $P$ of primes, weconsider the pseudovariety $\mathbf{G}_P$ of all finite groups having order aproduct of primes in $P$.

On the closure of cyclic subgroups of a free group in pro-V topologies