Profinite topology

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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