Let G be a group endowed with its profinite topology,
then G is called product separable if the profinite
topology of G is Hausdorff and, whenever H1,
H2, [ctdot ], Hn
are finitely generated subgroups of G, then the product subset
H1H2 [ctdot ]
Hn is closed in G. In this paper,
we prove that if G=F×Z is the direct product
of a free group and an infinite cyclic group, then G is
product separable. As a consequence, we obtain the
result that if G is a generalized free product of two
cyclic groups amalgamating a common subgroup, then
G is also product separable. These results generalize the
theorems of M. Hall Jr. (who proved the
conclusion in the case of n=1, [3]), and
L. Ribes and P. Zalesskii (who proved the conclusion in the case
of that G is a finite extension of a free group, [6]).