There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this

**Theorem ?** Let $U\subset\mathbb{R}^{2}=\mathbb{C}$ be a connected open subset, then $U$ is conformally equivalent to $\mathbb{R}^{2}-\cup_{n\in \mathbb{N}}I_{n}$, where $I_{n}$ is either empty or an interval of the form $[z,z+x_{n}]$ where $z$ is a complex number and $x_{n}$ is a positive real number.

**conformally equivalent** means that there exists a $C^{1}$-diffeomorphism $f:U\rightarrow \mathbb{R}^{2}-\cup_{n\in \mathbb{N}}I_{n}$ such that $df_{u}$ preserves angles for any $u\in U$.

- I would like to know if the statement is correct ?
- Does it admit a generalization in dimension 3. I mean a (
**EDIT**Not necessarily conformal ?) classification of open connected subsets of $\mathbb{R}^{3}$