Lemma 87.4.3. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module. Let $N \subset M$ be a submodule. Then
$0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge $ is exact, and
$N^\wedge $ is the closure of the image of $N \to M^\wedge $.
Proof. Let $M_\lambda $, $\lambda \in \Lambda $ be a fundamental system of open submodules. Then $N \cap M_\lambda $ is a fundamental system of open submodules of $N$ and $M_\lambda + N/N$ is a fundamental system of open submodules of $M/N$. Thus we see that (1) follows from the exactness of the sequences
and the fact that taking limits commutes with limits. The second statement follows from this and the fact that $N \to N^\wedge $ has dense image and that the kernel of $M^\wedge \to (M/N)^\wedge $ is closed. $\square$
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