Hello, There is a definition of simple-connectedness for a region R of the complex plane C that states that a region R is simply-connected in C if the complement of the region in the Riemann Sphere is connected. I don't know if I'm missing something; I guess we are actually consider the image of R under the stereo. projection,h, and defining connectedness using the topology of the 1-pt compactification of the plane. But it is not clear whether the image contains the point-at-infinity. So, say we have a strip S:=a< Imz <b , where Imz is the imaginary part of z. Then its image in the sphere would be a vertical strip going through the north pole N; but it seems like the image h(S) is connected if h(S) contains N, but not otherwise. So, what do we then do?