Inverse automata and profinite topologies on a free group

This paper gives an elementary, self-contained proof that a finite product of finitely generated subgroups of a free group is closed in the profinite topology. The proof uses inverse automata (graph immersions) and inverse monoid theory. Generalizations are given to other topologies. In particular, we obtain the new result that, for arborescent pseudovarieties, the product of two closed finitely generated subgroups is again closed. An application to monoid theory is given.