False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28

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.

FALSE: the Riemann sphere is homeomorphic to the complex plane

Statement

The Riemann sphere is homeomorphic to the complex plane.

Facts & Assumptions

Given: The Riemann sphere $\hat{C}$ and the complex plane $C$ .

[L1]

The one-point compactification of $C$ is compact and stereographic projection identifies it with the unit sphere ( $X^{*}$ 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, Stereographic projection identifies the Riemann sphere with the unit two-sphere).

Refutation

technique · direct

1.1

Fact [L1] makes $\hat{C}$ compact, while the open cover ${B (0, n) : n \geq 1}$ of $C$ has no finite subcover, so $C$ is not compact.

L1given

2.1

Homeomorphisms preserve compactness, so a compact space cannot be homeomorphic to a noncompact one. Hence the statement is false.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)