False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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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FALSE: a connected plane-domain complement in C already implies simple connectivity

Statement

False claim. If $Ω \subseteq C$ is a connected complex domain and $C ∖ Ω$ is connected, then $Ω$ is simply connected.

Facts & Assumptions

Given: The punctured plane witness from The punctured plane has connected complement in C but disconnected spherical complement.

[L1]

The punctured plane has connected complement in $C$ but disconnected spherical complement, and is therefore not simply connected (The punctured plane has connected complement in C but disconnected spherical complement).

Refutation

technique · direct

1.1

Apply [L1] to $Ω = C^{\times}$ . Its complement in $C$ is the connected set ${0}$ , but the domain is not simply connected.

givenL1

2.1

So the displayed implication fails. The missing point is exactly that plane simple connectivity depends on the complement in $\hat{C}$ , not merely in $C$ .

step 1.1∎

Depends on

The punctured plane has connected complement in C but disconnected spherical complement

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

J. Lebl, Guide to Cultivating Complex Analysis, Proposition 4.3.7 (standard reference, not scraped)