Riemann Surfaces
 Definition: A Riemann Surface $R$ is a second countable, connected, and Hausdorff topological space together with a collection of pairs $\{ (U_{\alpha}, \phi_{\alpha}) : \alpha \in \Gamma \}$ called an Atlas for $R$ with the following properties: 1) For each $\alpha \in \Gamma$, $U_{\alpha}$ is an open subset of $R$. 2) For each $\alpha \in \Gamma$, $\phi_{\alpha}$ is a homeomorphism of $U_{\alpha}$ into an open subset $V_{\alpha}$ of $\mathbb{C}$. 3) $\displaystyle{R = \bigcup_{\alpha \in \Gamma} U_{\alpha}}$. 4) For all $\alpha, \beta \in \Gamma$, the map $\phi_{\beta} \circ \phi_{\alpha}^{-1}$ is complex analytic from $\phi_{\alpha} (U_{\alpha} \cap U_{\beta})$ to $\phi_{\beta} (U_{\alpha} \cap U_{\beta})$.
2. A topological space is Hausdorff if for all $x, y \in R$ there exists open sets $U_x, U_y \subseteq R$ such that $x \in U_x$, $y \in U_y$, and $U_x \cap U_y = \emptyset$.
3. $\phi_{\alpha} : U_{\alpha} \to V_{\alpha}$ is a homeomorphism if $\phi_{\alpha}$ is a bijection where $\phi_{\alpha}$ is continuous on $U_{\alpha}$ and $\phi_{\alpha}^{-1}$ is continuous on $V_{\alpha}$.