Alphabeta MathHow 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.
Positive-codimension immersed submanifolds are null
Statement
Assume the Axiom of Countable Choice. Every immersed submanifold of positive codimension in a smooth manifold is a null subset of the ambient manifold.
Facts & Assumptions
Given: Countable Choice and an immersed -dimensional submanifold of an -manifold with .
An immersed submanifold is an -manifold equipped with a smooth injective immersion into the ambient manifold (Immersed submanifolds).
The image of a lower-dimensional manifold is null (The image of a lower-dimensional manifold is null).
Proof
By [F1], the immersed submanifold has a smooth injective immersion ; its image is the subset of in the statement.
Since , [L1] under the stated Countable Choice makes null in .
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
15 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
- Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)