Lemma 87.4.4. Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized $R$-module. Let $N \subset M$ be a closed submodule. If $M$ has a countable fundamental system of neighbourhoods of $0$, then $M/N$ is complete and the map $M \to M/N$ is open.
Proof. Let $M_ n$, $n \in \mathbf{N}$ be a fundamental system of open submodules of $M$. We may assume $M_{n + 1} \subset M_ n$ for all $n$. The system $(M_ n + N)/N$ is a fundamental system in $M/N$. Hence we have to show that $M/N = \mathop{\mathrm{lim}}\nolimits M/(M_ n + N)$. Consider the short exact sequences
\[ 0 \to N/N \cap M_ n \to M/M_ n \to M/(M_ n + N) \to 0 \]
Since the transition maps of the system $\{ N/N\cap M_ n\} $ are surjective we see that $M = \mathop{\mathrm{lim}}\nolimits M/M_ n$ (by completeness of $M$) surjects onto $\mathop{\mathrm{lim}}\nolimits M/(M_ n + N)$ by Algebra, Lemma 10.86.4. As $N$ is closed we see that the kernel of $M \to \mathop{\mathrm{lim}}\nolimits M/(M_ n + N)$ is $N$ (see Lemma 87.4.2). Finally, $M \to M/N$ is open by definition of the quotient topology. $\square$
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