0

An open connected set $S$ in $\mathbb{R}^2$ is said to be simply connected if its complement relative to the whole plane is connected. This definition is mentioned as an equivalent definition in the book by Tom Apostol Vol II, page 384.

With the above definition, it seems the set $\mathbb{R}^2\setminus\{0\}$ is simply connected, since its complement relative to the whole plane is the singleton set $\{0\}$. But with the other definition (Simply connected: An open and connected set $S$ in $\mathbb{R}^2$ is simply connected if for every simple closed curve $C$ which lies in $S$, the inner region of $C$ is also a subset of $S$) it is not simply connected.

Mathguide
  • 491
  • Maybe is there another hypothesis y you should add: relatively compact ? Otherwise @geetha290krm ‘s definition works. You can see this extended plane as the compactification of $\mathbb{R}^2$ – julio_es_sui_glace Apr 13 '24 at 09:06

1 Answers1

1

"An open connected set $S$ in $\mathbb{R}^2$ is said to be simply connected if its complement relative to the whole plane is connected". This is not the definition. You have to take the complement in the extended complex plane which is $\mathbb C \cup \{\infty\}$.

The complement of $\mathbb C \setminus \{0\}$ in the extended plane is $\{0,\infty\}$ which is not connected.

Ref. for the defintion: Characterizations of simple connectivity of subsets of $\mathbb{C}$

Kavi Rama Murthy
  • 359,332
  • Thanks a lot. I think it should be $\mathbb{R}^2\cup{\infty}$, but it is only mentioned relative to the whole plane. That's why I was confused. Thanks again. – Mathguide Apr 13 '24 at 09:37