Riemann surfaces and holomorphic atlases

Throughout, $\mathbb{C}$ carries its usual topology, and an open subset of $\mathbb{C}$ is a plane domain when it is nonempty and connected. A chart on a topological space $X$ is a homeomorphism $\phi :U\to \phi (U)$ from an open set $U\subseteq X$ onto an open subset $\phi (U)\subseteq \mathbb{C}$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); here $U$ is the domain of the chart and $\phi$ its coordinate. Two charts $\phi :U\to \mathbb{C}$ and $\psi :V\to \mathbb{C}$ are compatible when the transition maps $\psi \circ {\phi }^{-1}:\phi (U\cap V)\to \psi (U\cap V),\phi \circ {\psi }^{-1}:\psi (U\cap V)\to \phi (U\cap V)$ are holomorphic on the (possibly empty) open sets where they are defined.

A holomorphic atlas on $X$ is a family $\mathcal{A}$ of pairwise compatible charts whose domains cover $X$. A Riemann surface is a topological space $X$ such that

1. $X$ is nonempty and connected;
2. $X$ is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and second countable (Second countability: an at most countable basis for the topology);
3. $X$ carries a holomorphic atlas $\mathcal{A}$.

The three topological conditions say exactly that $X$ is a nonempty connected topological $2$-manifold without boundary (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces): each chart is a homeomorphism onto an open subset of $\mathbb{C}={\mathbb{R}}^2$, and conversely the local Euclidean condition is a supply of charts. The one-dimensional complex structure is the additional datum $\mathcal{A}$.

Maximal atlases. Fix a topological space $X$ and an atlas $\mathcal{A}$. A chart $\theta$ on $X$ is compatible with $\mathcal{A}$ when it is compatible with every member of $\mathcal{A}$. Write ${\mathcal{A}}^{\max }:=\{\theta :\theta \text{ is a chart on }X\text{ compatible with }\mathcal{A}\}\mathrm{.}$ Then $\mathcal{A}\subseteq {\mathcal{A}}^{\max }$, the family ${\mathcal{A}}^{\max }$ is an atlas, any two of its members are compatible, and it contains every atlas consisting of charts compatible with $\mathcal{A}$; in particular every holomorphic atlas on $X$ is contained in a unique maximal atlas, namely ${\mathcal{A}}^{\max }$. To see pairwise compatibility, let $\phi ,\psi$ belong to ${\mathcal{A}}^{\max }$ and let $p$ lie in their overlap. Since $\mathcal{A}$ covers $X$, there is a chart $\theta \in \mathcal{A}$ whose domain contains $p$. On this triple overlap, $\psi \circ {\phi }^{-1}=(\psi \circ {\theta }^{-1})\circ (\theta \circ {\phi }^{-1})$ is holomorphic as a composition of holomorphic maps, and the same holds for its inverse. Such neighbourhoods cover the overlap, proving compatibility. Any atlas containing $\mathcal{A}$ consists of charts compatible with $\mathcal{A}$, hence is contained in ${\mathcal{A}}^{\max }$; maximality therefore forces equality. The covering hypothesis is essential: a noncovering compatible family need not determine a unique complex structure on all of $X$. Thus "the complex structure of $X$" may be named by any one of its atlases, and two atlases determine the same complex structure exactly when their union is again an atlas.

Conventions fixed for this page and its companion.

• Nonemptiness is part of the definition. The empty space carries the empty atlas and is a $2$-manifold in the topological sense, but it is not a Riemann surface here; statements about Riemann surfaces may therefore use points.
• Connectedness is part of the definition. A disjoint union of two Riemann surfaces is a topological $2$-manifold with a holomorphic atlas but is not a Riemann surface; when a construction produces a possibly disconnected complex curve, its connected components are Riemann surfaces by restriction of the atlas.
• Charts are homeomorphisms. A chart is required to be a homeomorphism onto an open subset of $\mathbb{C}$; a merely holomorphic bijection onto a nonopen image would not be a chart. Because each transition is holomorphic with nowhere-vanishing derivative on a plane domain, $(\psi \circ {\phi }^{-1}{)}^{\mathrm{\prime }}$ has no zero and $\phi \circ {\psi }^{-1}$ is its holomorphic inverse; requiring holomorphy in both directions is therefore a symmetric formulation of the usual one-sided requirement, and it is kept because it is the form used here.
• The Riemann sphere and other examples. The standard two-chart atlas of the Riemann sphere, the identity atlas on a plane domain, and the quotient atlases of complex tori are produced on the companion page; this definition only fixes the axioms they must satisfy.
• No choice principle is used in the definition. An atlas is a set of charts; nothing selects a chart at a point, and a maximal atlas is determined by a first-order condition on charts.