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Riemann Surfaces (Definition)

One-dimensional complex manifolds. They are locally like the complex plane.
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The statement of the theorem

Let RR be a Hausdorff topological space. A Riemann surface is a structure (R,A)(R, \mathcal{A}) such that A={(Ui,ϕi)}i∈I\mathcal{A} = \{(U_i, \phi_i)\}_{i \in I} is an atlas satisfying the following conditions:\n\n1. **Covering:** The collection of open sets {Ui}i∈I\{U_i\}_{i \in I} covers RR (i.e., R=⋃i∈IUiR = \bigcup_{i \in I} U_i).\n2. **Local Homeomorphism:** For each i∈Ii \in I, ϕi:Ui→Vi⊂C\phi_i: U_i \to V_i \subset \mathbb{C} is a homeomorphism, where ViV_i is an open subset of C\mathbb{C}.\n3. **Holomorphic Compatibility:** For any pair of indices i,j∈Ii, j \in I such that Ui∩Uj≠∅U_i \cap U_j \neq \emptyset, the transition map ψji=ϕj∘ϕi−1\psi_{ji} = \phi_j \circ \phi_i^{-1} restricted to ϕi(Ui∩Uj)\phi_i(U_i \cap U_j) must be a biholomorphic map from ϕi(Ui∩Uj)\phi_i(U_i \cap U_j) to ϕj(Ui∩Uj)\phi_j(U_i \cap U_j).\n\nEquivalently, RR is a one-dimensional complex manifold whose transition functions are holomorphic.