Interior point

A point of a set with some whole neighborhood also inside the set.

A point $z_0$ is an interior point of a set $S$ if some neighborhood of $z_0$ , however small, lies entirely within $S$ . Interior points are the ones with breathing room: every direction you move a short enough distance, you're still in $S$ .

Examples

Every point of the open disk $|z|<1$ is an interior point of that disk.
In the closed disk $|z|\le 1$ , the point $z=1$ is not an interior point, since every neighborhood of $1$ contains points with $|z|>1$ .