Wikipedia: Pro-finite group

Difference (from prior major revision) (minor diff, author diff)

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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.

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.