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.