False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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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FALSE: the union of two domains of holomorphy is always a domain of holomorphy

Statement

The union of two domains of holomorphy is always a domain of holomorphy.

Facts & Assumptions

Given: The domains $U_{1} = C^{2} ∖ {z_{1} = 0}, U_{2} = C^{2} ∖ {z_{2} = 0} .$

[L1]

A domain of holomorphy is characterized by the impossibility of extending every holomorphic function across one common larger neighborhood (Holomorphic extension and domains of holomorphy in several variables).

[L2]

A holomorphic identity on a nonempty open subset propagates across a connected domain (A holomorphic function vanishing on a nonempty open subset of a domain vanishes identically).

[L3]

In complex dimension at least two, a holomorphic function on a punctured domain extends across the missing point (An isolated puncture is removable in complex dimension at least two).

Refutation

technique · direct

1.1

The function $1 / z_{1}$ is holomorphic on $U_{1}$ . If it extended across a point of the missing hyperplane ${z_{1} = 0}$ , then $z_{1} F$ would be holomorphic there and equal to $1$ on a nonempty open subset of the extension domain, so [L2] would force $z_{1} F \equiv 1$ , impossible where $z_{1} = 0$ . The same argument with $1 / z_{2}$ shows that $U_{2}$ is also a domain of holomorphy.

L1L2given

2.1

The union $U_{1} \cup U_{2}$ is exactly $C^{2} ∖ {(0, 0)}$ . By [L3], every holomorphic function on this punctured space extends across the origin, so [L1] shows that $U_{1} \cup U_{2}$ is not a domain of holomorphy. Thus the statement is false.

L1L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Jiří Lebl, Tasty Bits of Several Complex Variables, §1.6 and §2.1 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.3 (standard reference, not scraped)