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.

Every pro-finite group is totally disconnected and even more: a topological group is pro-finite if and only if it is Hausdorff, compact and totally disconnected.

Important examples of pro-finite groups are the p-adic integers. The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are pro-finite. In algebraic geometry the theory of the fundamental group is also a theory of pro-finite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety.