Simple-Connectedness in Complex Plane: Def. in Terms of Riemann Sphere.

[WWGD]

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?

 

[lavinia]

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?

The com[plement of a strip in the complex plane is two regions, the domains above and below the strip. This is not conected. But on the Riemann sphere they are joined at the north pole. This is connected.