Connected set

A set that is all one piece -- it cannot be split into two separated nonempty parts.

A set is connected if it forms a single piece: it cannot be split into two nonempty parts with no boundary contact between them. A practical test for the open sets used throughout this course is path-connectedness -- any two points of the set can be joined by a path that never leaves the set -- which agrees with connectedness for every open set that comes up here.

Examples

The disk $|z|<1$ is connected -- any two points inside it are joined by a straight segment that stays inside.
The union of two disjoint disks, $|z|<1$ together with $|z-10|<1$ , is not connected: no path from one piece to the other avoids leaving the set.