CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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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 $n\ge2$ , the sphere $S^{n-1}$ is path-connected and connected

Statement

For $n\ge2$ , the unit sphere $S^{n-1}\subseteq\mathbb R^n$ is path-connected and connected.

Facts & Assumptions

Given: $n\ge2$ and points $x,y\in S^{n-1}$ .

[L1]

The punctured space is polygonally connected, hence there is a continuous path in it from $x$ to $y$ (For $n\ge2$ , the punctured space $\mathbb{R}^n\setminus\{0\}$ is polygonally connected, A finite concatenation of straight segments in $\mathbb{R}^n$ is a continuous path).

[L2]

Radial normalisation is continuous on the punctured space and maps into the unit sphere (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 $\mathbb{R}^n$ ).

[L3]

A path-connected space is connected (Every path-connected space is connected, and every path component lies inside a component).

Proof

technique · constructive

1.1

Choose a path $\gamma:[0,1]\to\mathbb R^n\setminus\{0\}$ from $x$ to $y$ as in [L1], and define $\eta:=\rho\circ\gamma$ , where $\rho(z)=z/\lVert z\rVert_2$ .

L1L2construct

2.1

The map $\eta$ is continuous and takes values in $S^{n-1}$ by [L2]. Since $\rho(x)=x$ and $\rho(y)=y$ , it joins $x$ to $y$ in the sphere.

L2step 1.1

3.1

Thus the sphere is path-connected, and it is connected by [L3].

L3step 2.1discharge-construct∎

Depends on

Used by

FALSE: every connected subset of ℝⁿ is polygonally connected False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 107 results over 20 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

Sphere (standard reference, not scraped)
Path-connected space (standard reference, not scraped)

For n≥2n\ge2n≥2, the sphere Sn−1S^{n-1}Sn−1 is path-connected and connected