Sigma-compact open subgroup of a locally compact group
= 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 $\bigcup_{n\geq1}S^n$ 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.