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Let $H$ be a $\sigma-$compact, closed subgroup of locally compact abelian group $G$ such that $G/H$ is $\sigma-$compact. Is $G$ $\sigma-$compact?

Aliakbar
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    Yes. first note that Every closed subspace of a locally compact space is locally compact and If $G$ is locally compact then $G/H$ is locally compact. now use this theorem: If both $H$ and $G/H$ are locally compact and $σ$-compact then $G$ is locally compact and $σ$-compact. – M.Sina Aug 09 '13 at 11:31

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