Let $\mathcal{O}$ denote the class of open subsets of $\mathbb{C}$. For $U,V\in\mathcal{O}$, define $U\sim V$ if and only if there exists a bijective map $f:U\to V$ such that $f$ is holomorphic on $U$ and $f'(z)\neq 0$ for every $z\in U$; such a map is called a conformal isomorphism from $U$ to $V$. Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $\mathcal{O}$.