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 interior of an attached cell embeds openly in its closure
Statement
For an attached cell, its characteristic map is injective on and maps this interior homeomorphically onto an open subset of its closed-cell image. The full disk map need not be injective.
Facts & Assumptions
Given: The adjunction quotient of Cell attachment by a characteristic map.
Proof
The defining equivalence relation identifies only boundary points of with points of (and may identify boundary points with one another); no point of is equivalent to a distinct point.
The saturated open set has quotient homeomorphic to its image, and its complement in the closed-cell image is the image of . Thus the image is open relative to the attached space, while step 1.1 deliberately says nothing about the boundary.