RemarkRemark: AI-adaptedProof: Not applicablePipeline-generatedaudited 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.

The Riemann sphere is the published one-point compactification of the complex plane

Remark

Throughout this page write $\hat{C} : = C \cup {\infty} .$ By The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ , this is the one-point compactification of $C$ , and $X^{*}$ 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 gives the two facts used repeatedly below: $C$ sits inside $\hat{C}$ as an open dense subspace, and $\hat{C}$ is compact Hausdorff.

Nothing on this page redefines the underlying topological space. The new work is chartwise holomorphy at $\infty$ , the chordal metric, Möbius geometry, and the rational-map consequences built on that compactification.

The Riemann sphere is the published one-point compactification of the complex plane

Remark

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