ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Atlases on the sphere, plane, disc and annulus

Example

The two standard charts $ϕ_{0} (z) = z$ on $\hat{C} ∖ {\infty}$ and $ϕ_{\infty} (z) = 1 / z$ on $\hat{C} ∖ {0}$ , whose transition on the overlap $C^{\times}$ is $w \mapsto 1 / w$ , form a holomorphic atlas on the Riemann sphere; and for every nonempty connected open $Ω \subseteq C$ — in particular $Ω = C$ , the unit disc $D = {z : ∣ z ∣ < 1}$ , and every round annulus $A (0; r, R) = {z : r < ∣ z ∣ < R}$ with $0 < r < R$ — the single identity chart $i d : Ω \to C$ is a holomorphic atlas. Each of these examples satisfies all the Riemann-surface axioms of Riemann surfaces and holomorphic atlases, and the sphere's chart transition is $1 / z$ on $C^{\times}$ .

Facts & Assumptions

Given: The Riemann sphere $\hat{C}$ with its two standard charts, and the plane domains $C$ , $D$ and $A (0; r, R)$ .

[F1]

A Riemann surface is a nonempty connected Hausdorff second-countable space with a holomorphic atlas: charts are homeomorphisms onto open subsets of $C$ and compatible charts have holomorphic transitions in both directions (Riemann surfaces and holomorphic atlases).

[F2]

On $\hat{C}$ the sets $U_{0} = \hat{C} ∖ {\infty}$ and $U_{\infty} = \hat{C} ∖ {0}$ carry the charts $ϕ_{0} (z) = z$ and $ϕ_{\infty} (z) = 1 / z$ (with $ϕ_{\infty} (\infty) = 0$ ), and the transition maps on the overlap $C^{\times}$ are $w \mapsto 1 / w$ in both directions (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity).

[F3]

By The Riemann sphere is the published one-point compactification of the complex plane, $\hat{C}$ is the one-point compactification of $C$ ; hence it is compact and Hausdorff, $C$ is an open subspace and, $C$ being noncompact, $C$ is dense in $\hat{C}$ ( $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

[F4]

Stereographic projection is a homeomorphism $\hat{C} \to S^{2}$ , and $S^{2}$ is a subspace of $R^{3}$ (Stereographic projection identifies the Riemann sphere with the unit two-sphere).

[F5]

Every Euclidean space $R^{n}$ and every open subset of it is a smooth $n$ -manifold; a topological $n$ -manifold is by definition Hausdorff and second countable (Euclidean spaces and Euclidean open subsets as smooth manifolds, Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces).

[F6]

Both the Hausdorff property and second countability pass to subspaces ( $T_{0}$ , $T_{1}$ , and Hausdorffness are hereditary, Second countability is hereditary); a countable base of a space carries over to a countable base of any homeomorphic space.

[F7]

A path-connected space is connected; every convex subset of $R^{n}$ is path-connected (Every path-connected space is connected, and every path component lies inside a component, Every convex subset of $R^{n}$ , in particular every ball and $R^{n}$ itself, is path-connected and hence connected); the annulus is $A (0; r, R) = {z : r < ∣ z ∣ < R}$ with $0 < r < R$ (Annuli in the complex plane).

Verification

technique · direct

1.1F1F5F6given

(Plane domains give Riemann surfaces through the identity chart.) Let $Ω \subseteq C$ be nonempty, connected and open; as a subset of $R^{2}$ it is Hausdorff and second countable by [F5] and [F6], and the one-chart atlas ${i d : Ω \to C}$ has the holomorphic transition $i d \circ {i d}^{- 1} = i d$ on $Ω$ ; hence $Ω$ is a Riemann surface by [F1].

1.2F1F2F3F4F5F6given

(The sphere's two charts are a holomorphic atlas.) The domains $U_{0}, U_{\infty}$ cover $\hat{C}$ and each chart is a homeomorphism onto $C$ ; on the overlap the two transitions are $w \mapsto 1 / w$ , holomorphic on $C^{\times}$ [F2]; $\hat{C}$ is nonempty and, by [F3], compact Hausdorff, and it is connected because a separation of $\hat{C}$ would put the connected dense subspace $C$ on one side while the other side, being open and nonempty and containing a point of the closure of $C$ , meets $C$ ; it is second countable because $S^{2} \subseteq R^{3}$ is second countable by [F5] and [F6] and a homeomorphism transports a countable base, so [F4] transfers this to $\hat{C}$ ; hence $\hat{C}$ is a Riemann surface by [F1].

2.1F7step 1.1given

(The listed plane domains satisfy the hypotheses of step 1.1.) The plane $C$ and the unit disc $D$ are convex, hence path-connected and therefore connected by [F7]; the annulus $A (0; r, R)$ with $0 < r < R$ is nonempty: fix a finite radius $ρ \in (r, R)$ , taking $ρ = (r + R) / 2$ if $R < \infty$ and $ρ = r + 1$ if $R = \infty$ . It is path-connected because for $u \in A (0; r, R)$ the radial segment from $u$ to $\frac{ρ}{∣ u ∣} u$ stays in the annulus (its modulus runs between $∣ u ∣$ and $ρ$ , both in $(r, R)$ ) and any two points can be joined through the circle $∣ z ∣ = ρ$ ; therefore step 1.1 applies to $C$ , $D$ and $A (0; r, R)$ .

3.1step 1.1step 1.2step 2.1∎

(Conclusion.) Steps 1.1 and 2.1 exhibit the identity atlas on $C$ , on the unit disc and on every round annulus $A (0; r, R)$ with $0 < r < R$ , and step 1.2 exhibits the two-chart atlas on the sphere with transition $w \mapsto 1 / w$ on $C^{\times}$ ; in each case the transition maps are holomorphic and the Riemann-surface axioms hold, as claimed.

Remarks

The identity atlas on a plane domain is the smallest possible holomorphic atlas; its only transition is the identity. The sphere is the one case here where a second chart is needed; its standard two-chart atlas is contained in the maximal atlas of all charts compatible with it. The annulus is connected but not simply connected, and its identity chart is an injective coordinate onto the open annulus itself.

Depends on

Used by

The exponential map has no finite proper-map degree Counterexample
A nonsingular affine conic is a punctured-plane Riemann surface Example
Orders and residues under inversion on the sphere Example
Riemann–Hurwitz for the sphere power map Example

Dependency tree · two levels

56 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eduard Looijenga, Riemann Surfaces (2007) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Math 213b course notes (2026) (standard reference, not scraped)