Boundary point

A point where every neighborhood catches both points inside and outside the set.

A point $z_0$ is a boundary point of a set $S$ if every neighborhood of $z_0$ , no matter how small, contains at least one point in $S$ and at least one point not in $S$ . A boundary point need not itself belong to $S$ ; whether it does is exactly what separates an open set from a closed one.

Examples

Every point of the circle $|z|=1$ is a boundary point of the disk $|z|<1$ , even though none of them belong to that open disk.
The boundary of the half-plane $\operatorname{Re}(z)>0$ is the imaginary axis.