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