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. |
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. |