Pro-finite group

In mathematics, a pro-finite group G is a group that is the inverse limit of finite groups. Each of the finite groups is regarded as carrying the discrete topology, and since G is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group. Since all finite discrete spaces are compact Hausdorff spaces, their product will be a compact Hausdorff space by Tychonoff's theorem. G is a closed subset of this product and is therefore also compact Hausdorff. In fact, G is totally disconnected. Important examples of pro-finite groups are the p-adic integers.