# 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:

- 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 |