TITLE: compact inverse is compact in canonical homomorphism
QUESTION [3 upvotes]: Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there exists a compact $S\subset G$, such that $f(S)=C$.
Thanks for all the help

REPLY [2 votes]: The positive answer to your question is a corollary of Theorem 4.6.22 from “Topological groups and related structures” by Alexander V. Arhangel'skii and Mikhail G. Tkachenko.
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