Multiply connected domain

A domain with one or more holes -- some closed loop inside it cannot be shrunk to a point without leaving.

A domain is multiply connected if it is not simply connected: it has at least one hole, so some closed contour drawn inside it cannot be continuously shrunk to a point without crossing outside the domain. A domain with exactly one hole is doubly connected, with two holes triply connected, and so on. Cauchy's theorem fails outright on a multiply connected domain for a contour that circles a hole, which is why multiply connected domains need the separate machinery of contour deformation and the extended Cauchy integral formula.

Examples

The punctured disk $0<|z|<1$ is multiply (in fact, doubly) connected: a loop circling the missing center cannot shrink past it.
An annulus $1<|z|<2$ is doubly connected, with one hole -- the missing disk $|z|\le 1$ .