Punctured disk

A disk with its center removed: $0<|z-z_0|<r$ .

A punctured disk is an open disk $|z-z_0|<r$ with the single point $z_0$ deleted, written as the pair of inequalities $0<|z-z_0|<r$ . Removing just the center matters later on: limits and derivatives at $z_0$ only ever look at how a function behaves on points near $z_0$ , never at $z_0$ itself, and a punctured disk is exactly the shape of 'near, but not at.'

Examples

$0<|z|<1$ is the punctured unit disk -- every point of the open unit disk except the origin.
A function can be defined on the punctured disk $0<|z-2|<1$ while being left undefined at $z=2$ itself.