Equivalence of Definitions of Boundary

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

Definition $1$ is equivalent to Definition $3$
This is demonstrated in Boundary is Intersection of Closure with Closure of Complement.

Definition $2$ is equivalent to Definition $3$
Let $x \in S$.

By definition of the closure:
 * $x \in H^-$ every neighborhood $N$ of $x$ satisfies $H \cap N \ne \O$
 * $x \in \paren{\overline H}^-$ every neighborhood $N$ of $x$ satisfies $\overline H \cap N \ne \O$

Therefore $x \in H^- \cap \paren{\overline H}^-$ every neighborhood $N$ of $x$ satisfies:
 * $H \cap N \ne \O$

and
 * $\overline H \cap N \ne \O$