Alphabeta MathHow 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.
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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 By The one-point (Alexandroff) compactification , whose open sets are the open sets of together with the complements in of the closed compact subsets of , this is the one-point compactification of , and is compact and contains as an open subspace; is dense in exactly when is not compact; and is Hausdorff exactly when is locally compact and Hausdorff gives the two facts used repeatedly below: sits inside as an open dense subspace, and is compact Hausdorff.
Nothing on this page redefines the underlying topological space. The new work is chartwise holomorphy at , the chordal metric, Möbius geometry, and the rational-map consequences built on that compactification.
Depends on
- 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 $X$
- $X^{*}$ 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
Used by
- Circlines and their reflections on the Riemann sphere Definition
- Möbius transformations of the Riemann sphere Definition
- The cross-ratio of an ordered quadruple of sphere points Definition
- The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity Definition
- Stereographic projection identifies the Riemann sphere with the unit two-sphere Theorem
Dependency tree · two levels
15 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)