Definition 2

A topological group is said to be complete if its left and right uniformities are structures of complete spaces.

From Proposition 2 of no. 1, for a group to be complete it is sufficient that one of its uniformities is a structure of a complete space. $G$ is complete if and only if its associated Hausdorff group ($§\ 2$, no. 6) is complete.

Every closed subgroup of a complete group is complete (Chapter II, § 3, no. 4, Proposition 8). Every product of complete groups is complete (Chapter II, § 3, no. 5, Proposition 10).

On the other hand, if $G$ is a complete group and $H$ is a closed normal subgroup of $G$, then the quotient group $G/H$ is not necessarily complete (however, see Chapter IX, § 3, no. 1, Proposition 4).