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.
Every continuous map between smooth manifolds is homotopic to a smooth map
Statement
Every continuous map between smooth manifolds is homotopic to a smooth map.
Facts & Assumptions
Given: A continuous map between smooth manifolds.
Every continuous manifold-valued map admits a smooth approximation that is homotopic to it (Whitney approximation for manifold-valued maps).
Proof
Apply [L1] to the given continuous map and obtain a smooth map homotopic to it.
The map is smooth and lies in the homotopy class of the original map, so the claim follows.
Depends on
Used by
Dependency tree · two levels
6 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., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)