Region

An informal name for a subset of the complex plane, before any sharper label is attached.

A region is the everyday word this course uses for a subset of the complex plane under discussion -- a strip, a disk, a wedge, anything picked out by an equation or an inequality -- before more specific vocabulary like open, closed, connected, or domain gets attached to it. Calling something a region carries no promise about openness or connectedness on its own; those come later, once the sharper terms are available.

Examples

The set of $z$ with $1<|z|<2$ is a region -- an annulus -- long before we ask whether it is open or connected.
"Sketch the region where $\operatorname{Re}(z)>0$ " just means: draw the points satisfying that inequality.