I think we have restricted arg(z) to (pi/2,pi). After expanding z^2 = (e^i*theta)^2, you will observe that the argument changes to (pi,2pi), which indicates that the second quadrant (excluding axis) is getting mapped to the lower half of the complex plane (still excluding the axises). Thus you can easily prove that this set is open and connected. That is what I think about this question, correct me if I am wrong.