CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-07-31

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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For $n \geq 1$ , the punctured Euclidean space $R^{n} ∖ {0}$ is homotopy equivalent to $S^{n - 1}$

Statement

For every $n \geq 1$ , the punctured Euclidean space $R^{n} ∖ {0}$ and the unit sphere $S^{n - 1}$ have the same homotopy type.

Facts & Assumptions

Given: A natural $n \geq 1$ .

[L1]

The unit sphere is a deformation retract of $R^{n} ∖ {0}$ (For $n \geq 1$ , radial normalisation is a deformation retraction of $R^{n} ∖ {0}$ onto $S^{n - 1}$ ).

[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], $S^{n - 1}$ is a deformation retract of $R^{n} ∖ {0}$ .

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 · two levels

7 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

A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)
MAT 530 Topology lecture notes (Stony Brook University) (standard reference, not scraped)

For n≥1n≥1, the punctured Euclidean space Rn∖{0}Rn∖{0} is homotopy equivalent to Sn−1Sn−1