[Theorem 8.1, Ma]

Lemma 85.4.5. 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

1. $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0$ is exact,

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

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

Proof. We have $0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge$ is exact and statement (2) by Lemma 85.4.3. This produces a canonical map $c : M^\wedge /N^\wedge \to (M/N)^\wedge$. The module $M^\wedge /N^\wedge$ is complete and $M^\wedge \to M^\wedge /N^\wedge$ is open by Lemma 85.4.4. By the universal property of completion we obtain a canonical map $b : (M/N)^\wedge \to M^\wedge /N^\wedge$. Then $b$ and $c$ are mutually inverse as they are on a dense subset. $\square$

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