Profinite topology

Definition

The profinite topology on a group is a topology on the underlying set of the group defined in the following equivalent ways:

1. It has as a basis of open subsets all left cosets of subgroups of finite index.
2. It has as a basis of open subsets all right cosets of subgroups of finite index.
3. It has as a basis of open subsets all cosets of normal subgroups of finite index.

Under the profinite topology, any group becomes a topological group.

Group properties and the related properties of topological spaces

Group property	Property of topological space under profinite topology
finite group	discrete space
residually finite group	$T_{0}$-space; equivalently, $T_{1}$-space; equivalently, Hausdorff space, equivalently, regular space; equivalently, completely regular space. All these characterizations are equivalent for any topological group