Pro-finite group

A pro-finite group? is one which is the inverse limit of a finite? groups?. If we regard each of the finite? groups? as having the discrete topology, then as a subset of their product, it has a topology. Since all of the conditions on an inverse limit are closed in any Haussdorf? space, and since the product of compact? spaces is compact?, the inverse limit is compact? and Hausdorff? in the product topology.