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Let $M$ be a smooth properly embedded (closed) submanifold of $\Bbb R^n$ and suppose that $$\dim(M)<n.$$ Is it necessarily true that $\Bbb R^n-M$ is dense in $\Bbb R^n$?

Intuitively, it seems obviously true to me, but I cannot find a way to prove it.

user324991
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    Yes. You only have to prove that points of $M$ are limit points of sequences living outside of $M$. If $M = \mathbb R^d \varsubsetneq \mathbb R^n$, that's easy: $(x,0) = \lim (x,1/n)$. The general case reduces to this one through a coordinate chart. – PseudoNeo Mar 22 '16 at 10:49
  • @PseudoNeo Thanks, I understand that argument. – user324991 Mar 22 '16 at 11:00

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