Lemma 86.4.2. 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: