Pro-finite group

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.