Wikipedia: Pro-finite group

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of their product, the pro-finite group inherits 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, any pro-finite group is a compact Hausdorff space, and the group operations are continuous with respect to this topology.

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.

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.