LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.

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A homologically simply connected plane domain is either the plane or conformally equivalent to the disc

Statement

Assume the Axiom of Choice. Let $Ω \subseteq C$ be a homologically simply connected complex domain. Then either $Ω = C$ , or there is a biholomorphic map from $Ω$ onto the unit disc $D$ .

Facts & Assumptions

Given: The Axiom of Choice and a homologically simply connected complex domain $Ω$ .

[L1]

A homologically simply connected complex domain is, in particular, a complex domain (Homologically simply connected complex domains).

[L2]

Under the Axiom of Choice, every proper homologically simply connected complex domain is conformally equivalent to the unit disc (Every proper homologically simply connected plane domain is conformally equivalent to the unit disc).

Proof

technique · direct

1.1

By [L1], $Ω$ is a complex domain. If $Ω = C$ , then the first alternative holds and there is nothing more to prove.

givenL1

2.1

If $Ω \neq C$ , then $Ω ⊊ C$ is a proper homologically simply connected complex domain, so [L2] applies and gives a biholomorphic map $Ω \to D$ . This is exactly the second alternative.

step 1.1L2

3.1

Steps 1.1 and 2.1 prove the dichotomy.

step 1.1step 2.1∎

Depends on

Used by

For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent Theorem

Dependency tree · two levels

12 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 8 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Theorem 5.2.6 (standard reference, not scraped)