Hausdorff completion

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

The concept is named after Felix Hausdorff.