ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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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.
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The complex plane satisfies all of the grand-equivalent simple connectivity clauses

Example

The complex plane $C$ satisfies every clause of For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent.

Facts & Assumptions

Given: The complex plane $C$ .

[L1]

The grand theorem makes connected spherical complement, homological simple connectivity, trivial fundamental group, primitives, holomorphic logarithms, harmonic conjugates, conformal plane-or-disc alternative, and contractibility equivalent for plane domains (For a plane domain, the complement, homology, primitive, logarithm, conjugate, conformal, homotopy, and contractibility conditions are equivalent).

Verification

technique · direct

1.1

The spherical complement of $C$ is the singleton ${\infty}$ , hence connected.

given

2.1

By [L1], the connected complement from step 1.1 forces all remaining clauses. In particular, every cycle in $C$ is null-homologous, every entire function has a primitive, and $C$ is contractible.

step 1.1L1∎

Depends on

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

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4, §4.2 (standard reference, not scraped)