Lemma 15.36.4. With same assumptions as Lemma 15.36.3 if $M = \bigcup _{n \geq 1} N_ n$ for some closed subgroups $N_ n$, then $N_ n$ is open for some $n$.

Proof. If not, then $U_ n = M \setminus N_ n$ is dense for all $n$ and we get a contradiction with Lemma 15.36.3. $\square$

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