Lemma 87.4.2. Let $R$ be a topological ring. Let $M$ be a linearly topologized $R$-module and let $M_\lambda $, $\lambda \in \Lambda $ be a fundamental system of open submodules. Let $N \subset M$ be a submodule. The closure of $N$ is $\bigcap _{\lambda \in \Lambda } (N + M_\lambda )$.

**Proof.**
Since each $N + M_\lambda $ is open, it is also closed. Hence the intersection is closed. If $x \in M$ is not in the closure of $N$, then $(x + M_\lambda ) \cap N = 0$ for some $\lambda $. Hence $x \not\in N + M_\lambda $. This proves the lemma.
$\square$

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