A region in the complex plane is a non - empty, open, connected set. The following are all regions.
Any open disc
Any open half plane
The complementof any closed disc
Any open annulus
Any open rectangle, triangle, pentagon or similar shape
Any open, connected set with a finite number of points excluded
Proving the last of these is a set is quite easy.
Ifis a region andthenis also a region.
Since R is a region it contains an open disc centred onsois non – empty. Alsois closed insois open andis open as the intersection of two open sets.
Suppose thatSinceandis a region, we can joinand by a pathinand the path also lies inif
If howeverlies onthen choose an open disc(possible since R is a region) and modifyinside this disc to avoid
The resulting path joinsandinsois connected andis a region.
We can apply this region recursively to exclude any finite number of points