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.