Locally compact group

A locally compact group is a topological group whose underlying space is a locally compact space. In Haar measure and Fourier transform theory one normally also assumes it is a Hausdorff space. Local compactness supplies compact neighbourhoods, not just compact sets which algebraically generate the group.

A compact identity neighbourhood $K$ produces a symmetric compact neighbourhood $S=K\cup K^{-1}\cup \{e\}$. The subgroup ${\bigcup }_{n\ge 1}{S}^n$ is open because it contains an identity neighbourhood, and is sigma-compact because each finite product $S^n$ is compact. This construction does not require the original group itself to be sigma-compact.