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

The concept is named after Felix Hausdorff.

See also