Locally compact group

A locally compact group is a topological group $G$ whose underlying space is locally compact. Thus multiplication $G\times G\to G$ and inversion $G\to G$ are continuous, and every point has a neighborhood with compact closure.

Because left and right translations are homeomorphisms, it is enough to check local compactness at the identity element. Lie groups, finite groups with the discrete topology, and the additive groups of $\mathbb R^n$ and of the $p$-adic integers are standard examples.