Lemma 87.4.5. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Assume $M$ has a countable fundamental system of neighbourhoods of $0$. Then

$0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact,

$N^\wedge $ is the closure of the image of $N \to M^\wedge $,

$M^\wedge \to (M/N)^\wedge $ is open.

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