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.
The subspace topology on a leaf is always its manifold topology
Statement
The subspace topology on a leaf is always its manifold topology.
Facts & Assumptions
Given: Use the irrational linear leaf with irrational .
Intrinsically, the leaf is diffeomorphic to .
Refutation
As an immersed manifold, the leaf carries the topology transported from [given] by its parametrization.
If the subspace topology from agreed with that intrinsic [given] topology, the parametrization would be a topological embedding. The leaf would then be an embedded submanifold of the torus.
But the same irrational leaf is dense and not embedded. Therefore its [given] subspace topology cannot equal its manifold topology.
Hence the statement is false. [given] ∎
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
10 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
- Will J. Merry, Differential Geometry (standard reference, not scraped)