Hausdorff completion

In algebra, the Hausdorff completion formula_1 of a group "G" with filtration formula_2 is the inverse limit formula_3 of the discrete group formula_4. A basic example is a profinite completion. The image of the canonical map formula_5 is a Hausdorff topological group and its kernel is the intersection of all formula_2: i.e., the closure of the identity element. The canonical homomorphism formula_7 is an isomorphism, where formula_8 is a graded module associated to the filtration.
The concept is named after Felix Hausdorff.