In algebra, the Hausdorff completion of a group G with filtration is the inverse limit of the discrete group . A basic example is a profinite completion. The image of the canonical map is a Hausdorff topological group and its kernel is the intersection of all : i.e., the closure of the identity element. The canonical homomorphism is an isomorphism, where is a graded module associated to the filtration.

The concept is named after Felix Hausdorff.

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