A
pro-finite group is a
group which is the
inverse limit of
finite groups. If we regard each of the finite
groups as having the discrete
topology, then as a subset
of their product, the pro-finite group inherits a topology. Since all of the conditions
on an inverse limit are closed in any
Hausdorff space, and since
the product of
compact spaces is compact, any pro-finite group is a compact Hausdorff space, and the group operations are continuous with respect to this topology.