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I was thinking of a way to prove this and I realised that for my approach the lemma from the title would be useful, and it´s an interesting question on its own. Obviously it is true if the manifold is $\mathbb{R}^n$, but I don´t see it in general. This is the precise question:

Let $M$ be a connected manifold and $X$ a compact inside it, then is the union of $X$ and all the relatively compact components of $M\setminus X$ compact?

Edit: After a few days (and asking a few people) I thought the question is more adequate for MO, and I asked it there, where it has already been solved.

Saúl RM
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  • I just voted to close because I found a dupe, but this was a fun question to work on, and I'm glad you got an answer on MO. 10/10, would attempt and upvote again. – Jacob Manaker Dec 06 '21 at 08:17

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