DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Cell attachment by a characteristic map

Definition

For a space $X$ , an attaching map $f : S^{n - 1} \to X$ , and $n \geq 1$ , attach an $n$ -cell by the pushout $X \cup_{f} D^{n} = (X ⊔ D^{n}) / (z \sim f (z)$ for $z \in S^{n - 1})$ . The quotient map restricted to $D^{n}$ is its characteristic map; its image is the closed cell and the image of ${\overset{˚}{D}}^{n}$ is the open cell. For $n = 0$ , use $S^{- 1} = \emptyset$ , so $X \cup_{f} D^{0} = X ⊔ {*}$ .

Used by

The closure of a CW cell need not be a closed ball Counterexample
CW complex with closure finiteness and weak topology Definition
The interior of an attached cell embeds openly in its closure Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Allen Hatcher, Algebraic Topology, Chapter 0 (standard reference, not scraped)