Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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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.
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The chordal metric induces the standard topology of the Riemann sphere

Statement

The chordal metric χ induces the standard topology of the Riemann sphere. Equivalently, the identity map from C^ with the one-point-compactification topology to C^ with the metric topology of χ is a homeomorphism.

Facts & Assumptions

Given: The chordal metric χ and stereographic projection Σ.

[L1]

Stereographic projection is a homeomorphism C^→S2 (Stereographic projection identifies the Riemann sphere with the unit two-sphere).

Proof

technique · direct
1.1

By definition, χ(p,q)=∥Σ(p)−Σ(q)∥2, so Σ is an isometry from (C^,χ) onto S2 with its Euclidean subspace metric.

given
2.1

The Euclidean subspace metric induces the usual topology of S2, and [L1] already identifies that topology with the one-point-compactification topology on C^. Therefore χ induces the same topology.

L1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources