False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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 proper embedding of an n-manifold lands in R^n

Statement

False claim: every smooth $n$ -manifold admits a proper embedding into $R^{n}$ .

Facts & Assumptions

Given: The circle $S^{1}$ .

[L1]

Every smooth manifold does embed properly in $R^{2 n + 1}$ (The weak Whitney proper embedding theorem).

Refutation

technique · direct

1.1

Suppose $S^{1}$ embedded in $R$ . Because $S^{1}$ is compact and connected, its image would be a compact connected subset of $R$ , hence a closed interval $[a, b]$ .

givenalgebra

2.1

Removing any point from $S^{1}$ leaves a connected space, but removing an interior point from $[a, b]$ disconnects it, while removing an endpoint leaves a noncompact interval. Therefore $S^{1}$ is not homeomorphic to any closed interval. This contradicts step 1.1.

step 1.1algebra

3.1

So the claim already fails in dimension $1$ . The honest general statement is the higher-dimensional existence theorem [L1], not an ambient-dimension-equality theorem.

L1step 2.1∎

Depends on

The weak Whitney proper embedding theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Theorem 6.15 (standard reference, not scraped)