Hausdorff completion

In algebra, the Hausdorff completion $$\widehat{G}$$ of a group G with filtration $$G_n$$ is the inverse limit $$\varprojlim G/G_n$$ of the discrete group $$G/G_n$$. A basic example is a profinite completion. The image of the canonical map $$G \to \widehat{G}$$ 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 $$\operatorname{gr}(G) \to \operatorname{gr}(\widehat{G})$$ is an isomorphism, where $$\operatorname{gr}(G)$$ is a graded module associated to the filtration.

The concept is named after Felix Hausdorff.