Suppose $M$ is a differential manifold, $S$ an embedded submanifold, prove $S$ is closed in some open set of $M$. And I think it still holds up when $S$ is an immersed submanifold.
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First, note that for every $p \in S$ you can find an open set $U_p \subseteq M$ such that $S \cap U_p$ is a closed subset of $U_p$. To see it, just take a slice chart of $S$ around $p$. Then, each $U_p \setminus (S \cap U_p)$ is an open subset of $U_p$ which is an open subset of $M$ and so itself is an open subset of $M$ and
$$ \left( \bigcup_{p \in S} U_p \right) \setminus S = \bigcup_{p \in S} U_p \setminus (S \cap U_p) $$
is an open subset of $M$ contained in the open subset $\bigcup_{p \in S} U_p$ which shows that $S$ is closed in $\bigcup_{p \in S} U_p$.
If $S$ is merely immersed, then this doesn't hold. Consider for example an irrational winding on a two-dimensional torus which is dense in any open ball.
levap
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1thanks! but $\bigcup_{p \in M} U_p$ is just $M$, and $S$ may be an open set in $M$, do you mean $p \in S$? – user360777 Oct 11 '16 at 10:13
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1@user360777: Arg, yep. Thanks! Corrected. – levap Oct 11 '16 at 10:15