# A question regarding the Riemann sphere

According to Wikipedia, in the complex plane, an open set, say $$X$$, is simply connected if both $$X$$ and its complement are connected on the Riemann sphere.

Consider the complement of the closed unit disk, this set is open, and both the complement and the set are connected (at least that's what I would assume); however, this set isn't simply connected.

My question, is it possible to construct the complement of the closed unit disk using two disjoint open sets, i.e., is $$\{\infty\}$$ considered an open set?

Help me to understand, using this definition, why the set is not simply connected, and perhaps the significance of using the Riemann sphere in defining simple connectedness.

• en.wikipedia.org/wiki/Simply_connected_space
– Seth
Feb 7 at 3:38
• @Javier 4th paragraph of "Definition and equivalent formulations"
– Seth
Feb 7 at 3:41

• The definition I'm referring to assumes that any point in the complement can be connected to infinity. The 'inside' of the annulus, i.e, in the complement (not the interior of the annulus), cannot 'connect to infinity'. See Definition $8.1$ of Complex Analysis by Bak and Newman for the precise definition as there is the possibility that my understanding of it may be incorrect.