False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

An embedded submanifold need not be open in the ambient manifold

Statement

False claim: every embedded submanifold is an open subset of the ambient manifold.

Facts & Assumptions

Given: The $x$ -axis $S = {(x, 0) : x \in R} \subseteq R^{2}$ .

[F1]

Embedded submanifolds are locally coordinate slices (Embedded submanifolds and slice charts).

[F2]

Codimension records the dimension drop inside the ambient manifold (Codimension and hypersurfaces).

Refutation

technique · direct

1.1F1F2given

$S$ is the global coordinate slice $R \times {0}$ in $R^{2}$ , so [F1] makes it an embedded submanifold. Its codimension is $1$ by [F2].

2.1step 1.1

No Euclidean ball centred at a point of $S$ lies inside $S$ , because every such ball contains points with nonzero second coordinate. Hence $S$ is not open in $R^{2}$ .

3.1step 1.1step 2.1∎

Therefore an embedded submanifold need not be open in the ambient manifold.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Embedded Submanifolds (standard reference, not scraped)