The profinite topology on a group is a topology on the underlying set of the group defined in the following equivalent ways:
- It has as a basis of open subsets all left cosets of subgroups of finite index.
- It has as a basis of open subsets all right cosets of subgroups of finite index.
- 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 property||Property of topological space under profinite topology|
|finite group||discrete space|
|residually finite group||-space; equivalently, -space; equivalently, Hausdorff space, equivalently, regular space; equivalently, completely regular space. All these characterizations are equivalent for any topological group|