A set S is called star-like if there exists a point $\alpha\in S$ such that the line segment connecting $\alpha$ and z is contained in S for all $z\in S$. Show that a star-like region is simply connected.
My answer
Show that $γ:γ(t)=tz+(1−t)α, t≥1$ is contained in the complement for any z in the complement
Let $\gamma$ represents the portion of the ray from $\alpha$ through z to $\infty$, starting at z. Thus, if z is in the complement of S, so is all of $\gamma$. For, if any $z_{1}\in \gamma$ belonged to S, so would the entire segment connecting $\alpha$ and $z_{1}$, including z.
Could anyone help to formalize the answer?