This is a hard problem in my book ( marked with a star). I don't even see the
question in the problem. I guess it's the last sentence, but I still have no clue to
solve it. Hope someone can gives help.

Suppose $\displaystyle \Omega$ is a bounded region. Let $\displaystyle L $ be a (two way infinite) line that intersects $\displaystyle \Omega$. Assume that $\displaystyle \Omega\cap L$ is an interval $\displaystyle I$. Choosing an orientation for $\displaystyle L$, we can define $\displaystyle \Omega_l$ and $\displaystyle \Omega_r$ to be subregions of $\displaystyle \Omega$ lying strictly to the left or right of L, with $\displaystyle \Omega=\Omega_l \cup I \cup \Omega_r$ a disjoint union. If $\displaystyle \Omega_l$ a $\displaystyle \Omega_r$nd are simply connected, then $\displaystyle \Omega$ is simply connected.