Boundary

The set of all boundary points of $S$, denoted $\partial S$, is called the boundary of $S$.

A point $z_0$ is called a boundary point of $S$ if $\mathrm{\forall }ϵ>0$, $U_{ϵ}(z_0)\cap S\ne 0$ and $U_{ϵ}(z_0)\cap (\mathbb{C}/S)\ne 0$. That is to say, every $ϵ$-neighborhood of $z_0$ contains a point in $S$ and a point not in $S$. Note that a boundary point may or may not belong to the set.