TITLE: A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.
QUESTION [6 upvotes]: Let $E$ be a subset of a metric space $(S,d)$.
Prove that:
A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.

Here is what I thought:
I'm first trying to understand what I need to prove.
The boundary of $E$ is the set $E^- -E ^{\circ}$.
It a point belongs to both $E^-$ and $(S-E)^-$, then it belongs to $E^- \cap (S-E)^-$.
Therefore I think I need to prove that:
$$E^- -E ^{\circ}=E^- \cap (S-E)^-$$
And now I'm not sure if there any set theory rules I could use.
Intuitive I would say that I need to prove: $S-E ^\circ =(S-E)^-$. Is this correct ?
And I can kind of feel that this last statement is right, but I can't prove this rigoursly.
Could anybody help me how I can prove this statement more rigourisly ?

REPLY [2 votes]: Yes. $S-E^\circ=\overline{S-E}$ is correct.

$x\notin E^\circ \iff $ there is no whole ball around $x$ within $E$ $\iff$  all balls around $x$ intersect $S-E$ $\ \iff x\in\overline{S-E}$.