 Sigma-compact open subgroup of a locally compact 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.