False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 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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FALSE: every connected subset of $\mathbb{R}^n$ is polygonally connected

Statement

False claim: every connected subset of $\mathbb R^n$ is polygonally connected.

The unit circle $S^1\subseteq\mathbb R^2$ is connected but is not polygonally connected.

Facts & Assumptions

Given: The unit circle $S^1=\{u\in\mathbb R^2:\lVert u\rVert_2=1\}$ and the points $e_0,-e_0\in S^1$ .

[A1]

Every connected subset of Euclidean space is polygonally connected.

[L1]

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

[L2]

A polygonal path is a finite concatenation of straight segments (Polygonal paths and polygonally connected subsets of $\mathbb{R}^n$ ).

[L3]

If distinct unit vectors $u,v$ are joined by a segment, its midpoint has squared Euclidean norm $\lVert(u+v)/2\rVert_2^2=1-\lVert u-v\rVert_2^2/4<1$ (The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ , A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).

Refutation

technique · contradiction

1.1

Suppose the claim [A1] holds. Since $S^1$ is connected by [L1], there is a polygonal path in $S^1$ from $e_0$ to $-e_0$ .

A1L1assume-contra

2.1

In its finite vertex list some adjacent vertices $u,v$ are distinct, since its endpoints are distinct. The straight segment from $u$ to $v$ lies in $S^1$ by [L2].

L2step 1.1

3.1

But the midpoint of this segment has norm strictly less than $1$ by [L3], so it does not lie in $S^1$ . This contradicts step 2.1.

L3step 2.1discharge-contradiction∎

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: 115 results over 22 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)
Convex set (standard reference, not scraped)

FALSE: every connected subset of Rn\mathbb{R}^nRn is polygonally connected