Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31
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.

For n≥1n\ge1, the punctured Euclidean space Rn∖{0}\mathbb{R}^n\setminus\{0\} is homotopy equivalent to Sn−1S^{n-1}

Statement

For every n≥1n\ge1, the punctured Euclidean space Rn∖{0}\mathbb R^n\setminus\{0\} and the unit sphere Sn−1S^{n-1} have the same homotopy type.

Facts & Assumptions

Given: A natural n≥1n\ge1.

[L1]

The unit sphere is a deformation retract of Rn∖{0}\mathbb R^n\setminus\{0\} (For n≥1n\ge1, radial normalisation is a deformation retraction of Rn∖{0}\mathbb{R}^n\setminus\{0\} onto Sn−1S^{n-1}).

[L2]

The inclusion of a deformation retract is a homotopy equivalence (The inclusion of a deformation retract is a homotopy equivalence with the retraction as homotopy inverse).

Proof

technique · direct
1.1

By [L1], Sn−1S^{n-1} is a deformation retract of Rn∖{0}\mathbb R^n\setminus\{0\}.

L1
2.1

By [L2], its inclusion is a homotopy equivalence, so the two spaces have the same homotopy type.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources