TheoremStatement: Literature-sourcedProof: AI-adaptedSession-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 $n\ge1$ , radial normalisation is a deformation retraction of $\mathbb{R}^n\setminus\{0\}$ onto $S^{n-1}$

Statement

Let $n\ge1$ , put $P=\mathbb R^n\setminus\{0\}$ , and let $S^{n-1}\subseteq P$ be the unit sphere. Radial normalisation

$r:P\to S^{n-1},\qquad r(x)=\frac{x}{\lVert x\rVert_2},$

is a retraction, and

$H(x,t)=\left((1-t)+\frac{t}{\lVert x\rVert_2}\right)x$

is a deformation retraction of $P$ onto $S^{n-1}$ .

Facts & Assumptions

Given: A natural $n\ge1$ , $P=\mathbb R^n\setminus\{0\}$ and $S^{n-1}=\{x:\lVert x\rVert_2=1\}$ .

[L1]

Radial normalisation $r:P\to S^{n-1}$ is continuous (Radial normalisation $x\mapsto x/\lVert x\rVert_2$ is continuous on $\mathbb{R}^n\setminus\{0\}$ , Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$ ).

[L2]

The displayed $H:P\times I\to P$ is continuous, begins at $x$ , ends at $r(x)$ , fixes every $s\in S^{n-1}$ , and never reaches $0$ (For $n\ge1$ , the map $H(x,t)=((1-t)+t/\lVert x\rVert_2)x$ is continuous on $(\mathbb{R}^n\setminus\{0\})\times[0,1]$ , starts at $x$ , ends at radial normalisation, fixes the unit sphere, and never reaches $0$ ).

[A1]

A deformation retraction onto $A$ is a retraction $r$ together with a homotopy from the identity to the inclusion followed by $r$ , fixed pointwise on $A$ (Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise).

Proof

technique · direct

1.1

If $s\in S^{n-1}$ then $\lVert s\rVert_2=1$ , so $r(s)=s$ . Thus the continuous map $r$ of [L1] is a retraction.

L1algebra

1.2

By [L2], $H$ is a continuous homotopy in $P$ from $\operatorname{id}_P$ to the inclusion followed by $r$ , and $H(s,t)=s$ for every $s\in S^{n-1}$ and $t\in I$ .

2.1

Steps 1.1 and 1.2 satisfy [A1], so $(r,H)$ is a deformation retraction of $P$ onto $S^{n-1}$ .

step 1.1step 1.2A1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 19 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

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\ge1n≥1, radial normalisation is a deformation retraction of Rn∖{0}\mathbb{R}^n\setminus\{0\}Rn∖{0} onto Sn−1S^{n-1}Sn−1