CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A priori a locally compact topological group is a topological group $G$ whose underlying topological space is locally compact.
Typically it is also assumed that $G$ is Hausdorff. (Notice that if not, then $G/\overline{\{1\}}$ is Hausdorff.).
One often says just “locally compact group”.
We take here locally compact groups $G$ 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 $\rho$ and that $L^1(\rho)$ 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 $H$ of a locally compact topological group $G$ is itself locally compact (in induced topology) iff it is closed in $G$.
Again taking locally compact groups $G$ to be Hausdorff, such are complete both with respect to their left uniformity and their right uniformity. For if $\{x_\alpha\}$ is a Cauchy net in $G$ and $U$ is a compact neighborhood of the identity $e$, then there is $\alpha$ so large that $x_\beta x_\alpha^{-1} \in U$ for all $\beta \geq \alpha$. Those elements converge to a point $x \in U$ since $U$ is compact, and the original net converges to $x \cdot x_\alpha$. A similar argument is used for the right uniformity.