They ask us to show that the sets are open and connected while the solution only shows what the sets are.
Does it mean that in the quiz, we can just show what the set is and then say "thus it is open and connected"? (It's is obvious since it only involves rotation and shrinking and an offset).
Since I think I cannot write the proof down in time in a quiz. Because I have to find an appropriate radius for the ball which involves a complicated computation unless I can use some theorem like:
"a function is continuous if and only if its inverse mapping preserve open" and
In this question, a segment after $z \mapsto \alpha z + \beta$ mapping is a segment which means I can prove polygonal connectedness, while I cannot do the same thing in Q10 though the set is much simpler.
