A star shaped domain $U\subseteq \mathbb{C}$ is 1;;simply connected. $U$ is said to be star shaped with respect to $z_0\in U$ if for any $z_1\in U$, the segment between $z_0$ and $z_1$ is contained in $U$.

If we let $U=\mathbb{C}-\{0\}$, then for any $z_0\in U$, $\pi (U,z_0)$ is non-trivial.