Functions of a complex variable Definition

Riemann surface

Builds on Topological space Hausdorff space Connected space Homeomorphism Open cover Holomorphic function

Definition

Let $X$ be a topological space. A complex chart on $X$ is a homeomorphism $\varphi \colon U \to V$ from an open set $U \subseteq X$ onto an open set $V \subseteq \C$ . Two charts $\varphi_1 \colon U_1 \to V_1$ and $\varphi_2 \colon U_2 \to V_2$ are compatible if the transition map $\varphi_2 \circ \varphi_1^{-1} \colon \varphi_1(U_1 \cap U_2) \to \varphi_2(U_1 \cap U_2)$ is holomorphic, and likewise $\varphi_1 \circ \varphi_2^{-1}$ . A complex atlas on $X$ is a family of pairwise compatible charts whose domains form an open cover of $X$ . Two atlases are equivalent if their union is again an atlas. A complex structure on $X$ is an equivalence class of complex atlases. A Riemann surface is a topological space $X$ with a complex structure such that:

$X$ is Hausdorff, and
$X$ is connected.

Remarks

Some authors also require $X$ to be second countable; by Radó's theorem this holds automatically for every Riemann surface. So a Riemann surface is the same as a connected complex manifold of dimension one.

Example: the Riemann sphere $\hat\C = \C \cup \set{\infty}$ with the two charts $\varphi_1(z) = z$ on $\C$ and $\varphi_2(z) = 1/z$ on $\hat\C \setminus \set{0}$ , where $\varphi_2(\infty) = 0$ . On $\varphi_1(\C \setminus \set{0}) = \C \setminus \set{0}$ the transition map is $z \mapsto 1/z$ , which is holomorphic there; its inverse is the same map.

Non-example: the closed unit disc $\set{z : \abs{z} \le 1}$ is not a Riemann surface for any atlas. A boundary point has no open neighbourhood homeomorphic to an open subset of $\C$ , so the domains of charts cannot cover $X$ .

Reading path

To understand this page from the ground up, read these 51 pages in order. The foundations come first.

Complex numbers object
Element definition
Proposition concept
Real numbers object
Set bounded above definition
Real and imaginary parts definition
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Completeness axiom statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Existence of square roots of nonnegative reals statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Square root definition
Complement definition
Modulus of a complex number definition
Function definition
Complex differentiability definition
Preimage of a set definition
Metric space definition
Codomain definition
Injective function definition
Holomorphic function definition
Open ball definition
Surjective function definition
Metric topology definition
Bijective function definition
Inverse function definition
Open set definition
Closed set definition
Continuous map of topological spaces definition
Hausdorff space definition
Open cover definition
Homeomorphism definition
Connected space definition
Riemann surface definition

Tag	`WFREP1QRPQ`
URI	`m2://WFREP1QRPQ`
Kind	definition
Writing	W0 stub
MSC	30F10, 30F20, 14H55
Defines	Riemann surface
Uses	Topological space, Hausdorff space, Connected space, Homeomorphism, Open cover, Holomorphic function
Changed	2 November 2026

Riemann surface

Definition

Remarks

Reading path

Further reading

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