Typically it is also assumed that is Hausdorff. (Notice that if not, then is Hausdorff.).
One often says just “locally compact group”.
We take here locally compact groups to be also Hausdorff.
Locally compact topological groups are the standard object of study in classical abstract harmonic analysis. The crucial properties of locally compact groups is that they posses a left (right) Haar measure and that has a structure of a Banach -algebra.
A left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under the left (right) multiplications by elements in the group. A topological subgroup of a locally compact topological group is itself locally compact (in induced topology) iff it is closed in .
Again taking locally compact groups to be Hausdorff, such are complete both with respect to their left uniformity and their right uniformity. For if is a Cauchy net in and is a compact neighborhood of the identity , then there is so large that for all . Those elements converge to a point since is compact, and the original net converges to . A similar argument is used for the right uniformity.