09 - Types of Subsets

Convex

Let $X$ be a subset in $\mathbb{C}$ then

$X$ is convex if
For all $z,w\in \mathbf{x}$ then

$$\text{line segment between }z\text{ and }w\text{ is contained in }X$$

Star-Like

Let $X$ be a subset in $\mathbb{C}$ then

$X$ is star-like if
There exists point $z_0$ in $X$ such that for all $w\in X$ then

$$\text{line segment between }{z}_0\text{ and }w\text{ is contained in }X$$

so that $X$ is star-like with respect to $z_0$

Annulus

$$A(x_0,r,R)=\{z\in \mathbb{C}:0\le r<|z-z_0|<R\le \infty \}$$

Note that annulus is not simply connected