Embedded submanifold is closed in some open set of total manifold

Question

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.

Answer 1

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.