Show that the the definition of the boundary of a subset E

of metric space X, $partial S : = {p in X: forall r>0, E cap

> B_r(p) neq emptyset text{ and } E^C cap B_r(p) neq emptyset }$

is equivalent to the definition of the boundary in case of a set E in

the topological space $(X, mathcal{T})$, $partial S := overline{E} setminus E^0$

Attempted solution:

Let $C := {p in X: forall r>0, E cap B_r(p) neq emptyset text{ and } E^C cap B_r(p) neq emptyset }$ and $D := overline{E}setminus E^0$. In order to show $C = D$, we need to first show that for any $x in C text{ implies } x in D$ and then show that for any $x in D text{ implies } x in C$.

Let $x in C$. Then, $E cap B_r(x) neq emptyset$ and $E^C cap B_r(x) neq emptyset$. Consider an arbitrary set $F_alpha$ such that $F_alpha$ is a closed subset in $X$ and $E subseteq F_alpha$. Now , I need to show that $x in F_alpha$ , which implies that $x in bigcap_{alpha} F_alpha = overline{E}$, but I have no idea how to do this.

Next, I need to show that $x notin E^0$. I am assuming we do a proof by contradiction but I am not really sure how to proceed.

Also, I have no idea on how to proceed in the reverse directions. Can someone point me in the right direction? Thanks!