Hausdorff completion

In algebra, the Hausdorff completion G^ of a group G with filtration Gn is the inverse limit lim<-G/Gn of the discrete group G/Gn. 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 Gn: 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.

## See also

* Linear topology

## References

* Nicolas Bourbaki, Commutative algebra

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