Definition 4.5.9 (Boundary). Let $X$ be a topological space, $A \subset X$, $x \in X$, and $\fB \subset \cn(x)$ be a fundamental system of neighbourhoods, then the following are equivalent:
For every $V \in \cn(x)$, $V \cap A \ne \emptyset$ and $V \cap A^{c} \ne \emptyset$.
For every $V \in \fB$, $V \cap A \ne \emptyset$ and $V \cap A^{c} \ne \emptyset$.
$x \in \overline{A}\setminus A^{o}$.
$x \in \overline{A}\cap \overline{A^c}$.
The set $\partial A$ of all points satisfying the above is the boundary of $A$.
Proof. $(1) \Rightarrow (2)$: $\cn(x) \supset \fB$.
$(2) \Rightarrow (3)$: By (2) of Definition 4.5.2, $x \in \overline{A}$. Since $V \cap A^{c} \ne \emptyset$ for all $V \in \fB$, there exists no open set $U \subset A$ with $x \in A$. By (2) of Definition 4.5.1, $x \not\in A^{o}$.
$(3) \Rightarrow (4)$: By Lemma 4.5.8.
$(4) \Rightarrow (1)$: By (2) of Definition 4.5.2.$\square$