Is there a simply connected region with only two (or 1<N<$+\infty$) boundary points?

My textbook states the Riemann mapping theorem as follows:

If D is a simply-connected domain on the extended complex plane that has at least two boundary points ... (translated)

I'm wondering what a simply-connected region with only two boundary points would be.

Definition of simply-connected domain:

For every simple closed curve C in domain D, all points in the interior of C are also in D, where the "interior" means: a simple closed curve in the plane divides the plane into two regions, one exterior, one interior.

What I know:

1. There are both points belonging and not belonging to the set in any neighborhood of the boundary point.

2. The complement of a simply-connected region is a connected region.

PS. I major in physics. I don't know much about this problem and the translation maybe not very clear. The textbook is specially written for physics students as well.

Thanks a lot.

Solution 1:

No, there isn't. If the region has finitely many boundary boundary points, then the boundary is equal to the complement of the region. But

• The complement of a simply-connected region is connected.
• Connected metric spaces with at least 2 points are uncountable.

Solution 2:

There are no simple connected domains with 2 boundary points, but the complete extended complex plane is a simple connected domain without boundary points. The extended complex plane with one point removed is a simple connected domain with one boundary point. Those are all the examples. So my guess is that your textbook is using "with at least two boundary points" as a shortcut of "different from the whole extended complex plane or the extended complex plane with a point removed"