I come across this definition while studying complex analysis:
An (open) set $U \subset \mathbb C$ is star-shaped there exists a point $a_0$ such that for all $w\in U$ the line segment connecting $w$ and $a_0$ lies completely inside $U$.
With this definition, we proceed to prove the Cauchy-Goursat Theorem of Integrals for Star Domains:
If in a star-shaped domain $U$, we have a holomorphic function $f$ and a piecewise smooth closed curve $\gamma$, then $\int_\gamma f(z) dz = 0$.
And when I continue my studying, most of the application (such as solving integrals) relies on the fact that $U$ is a simply connected set. By reading this post, I also found that $U$ is star-shaped $\Rightarrow$ $U$ is simply connected $\Rightarrow$ $U$ is path-connected.
I also heard from others that the Cauchy-Goursat Theorem of Integrals for Star Domains is a theorem stronger than required. So does there exists a set such that
- $U$ is simply connected but not star-shaped?
- $U$ is path-connected but not star-shaped?
Thank you!
Half of an annulus will do the job. In terms of an equation, $$\{(x, y) \in \Bbb R^2 : 1 \leqslant x^2 + y^2 \leqslant 2, \ x \geqslant 0\}.$$ Do you see why this satisfies both your conditions?