CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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.
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For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected

Statement

For $n \geq 2$ , the unit sphere $S^{n - 1} \subseteq R^{n}$ is path-connected and connected.

Facts & Assumptions

Given: $n \geq 2$ 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 $n \geq 2$ , the punctured space $R^{n} ∖ {0}$ is polygonally connected, A finite concatenation of straight segments in $R^{n}$ is a continuous path).

[L2]

Radial normalisation is continuous on the punctured space and maps into the unit sphere (Radial normalisation $x \mapsto x / ∥ x ∥_{2}$ is continuous on $R^{n} ∖ {0}$ , Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

[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 $γ : [0, 1] \to R^{n} ∖ {0}$ from $x$ to $y$ as in [L1], and define $η : = ρ \circ γ$ , where $ρ (z) = z / ∥ z ∥_{2}$ .

L1L2construct

2.1

The map $η$ is continuous and takes values in $S^{n - 1}$ by [L2]. Since $ρ (x) = x$ and $ρ (y) = y$ , 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

A connected plane domain that is not homologically simply connected Counterexample
Antipodal points on a round sphere have many minimizing geodesics Counterexample
FALSE: every connected subset of ℝⁿ is polygonally connected False statement
The exterior of a closed disc in the plane is path-connected Lemma
The quaternion double cover generates the third homotopy group of SO(3) Lemma

Dependency tree · two levels

24 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

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

For n≥2n≥2, the sphere Sn−1Sn−1 is path-connected and connected