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.

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.