Open set

A set in which every point has some neighborhood entirely inside the set.

A set $S$ is open if every point of $S$ is an interior point of $S$ -- that is, around each $z_0\in S$ there is some neighborhood $|z-z_0|<\varepsilon$ that fits entirely inside $S$ , with no part of it poking outside. Informally, an open set has no edge belonging to it: you can always nudge a little in any direction and stay inside.

Examples

The open disk $|z|<1$ is open: shrink the neighborhood radius enough at any interior point and it stays inside the disk.
The half-plane $\operatorname{Re}(z)>0$ is open, but $\operatorname{Re}(z)\ge 0$ is not, since points on the imaginary axis have no neighborhood avoiding $\operatorname{Re}(z)<0$ .