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 .