4

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.

cineel
  • 1,547
  • 4
  • 11
  • 23
BetaGem
  • 41
  • How is simple connectivity defined in your textbook? Do you already know that $X={\mathbb C}\setminus {0}$ is not simply-connected? If not, what are other theorems about simply-connected domains that you know and that would show that $X$ is not simply-connected? – Moishe Kohan Dec 13 '21 at 08:49
  • @MoisheKohan It's defined as follows: for every simple closed curve $C$ in domain $D$, all points in the interior of $C$ are also in $D$. I know $X$ is not simply-connected if $z=\infty$ is not taken into account. I'm a physics major, and I don't know much about this. The textbook is specially written for physics students as well. – BetaGem Dec 13 '21 at 09:07
  • 2
    Ok, then edit the question to make it clear what you know and what you do not know. Include the definition of the "interior" of a simple closed curve. Incidentally, the definition they gave you is quite horrible since it hinges upon Jordan curve theorem, which is a nontrivial result. – Moishe Kohan Dec 13 '21 at 09:10
  • 1
    Which textbook are you referring to? – Shaun Dec 15 '21 at 16:00
  • 1
    @Shaun Functions of a Complex Variable (2nd Edition), Zhenjun Yan, Press of USTC. It's written in Chinese. (《复变函数》严镇军) – BetaGem Dec 16 '21 at 06:49

2 Answers2

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

Martin R
  • 128,226
  • 1
    Is something wrong here? Please let me know! – Martin R Dec 13 '21 at 08:48
  • Don't know, but maybe due to EoQS. Incidentally, it's unclear if OP knows why the first item holds (there are so many competing definitions of simple connectivity used in CA textbooks). – Moishe Kohan Dec 13 '21 at 09:00
  • Why is the boundary equal to the complement of the region if the region has finitely many boundary points? – BetaGem Dec 13 '21 at 10:46
  • 1
    @BetaGem: Otherwise the complement would contain an open disk, and then every line from the center of that disk to some point in the region would meet the boundary somewhere. – Martin R Dec 13 '21 at 10:48
  • @MartinR I still don't understand. What's the contradiction if the lines meet the boundary? – BetaGem Dec 13 '21 at 11:14
  • 1
    @BetaGem: Because you get infinitely many boundary points, one for every direction of such lines. – Martin R Dec 13 '21 at 11:17
  • @MartinR I see. It seems that I really lack knowledge in this field. Thank you. – BetaGem Dec 13 '21 at 11:30
  • @BetaGem: Which part is still unclear? – Martin R Dec 13 '21 at 12:07
  • The picture seems clear now, but I don't know how to express it strictly in mathematics. – BetaGem Dec 13 '21 at 14:28
1

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"

jjagmath
  • 22,582