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.

Skeleta, CW subcomplexes, and relative CW complexes

Definition

Write $X^{n}$ for the union of cells of dimension at most $n$ , with $X^{- 1} = \emptyset$ . A CW subcomplex $A \subseteq X$ is a union of open cells such that, whenever $A$ contains an open cell $e$ , it contains the whole closure $\overline{e}$ . A relative CW complex $(X, A)$ is formed from the subcomplex $A$ by attaching cells in stages; thus $A ↪ X$ is a cellular inclusion.

Depends on

CW complex with closure finiteness and weak topology

Used by

The image of a compact space lies in a finite CW subcomplex Corollary
Incidence number of two CW cells Definition
A CW complex is the colimit of its skeleta in the weak topology Proposition
Cellular maps induce cellular chain maps Proposition
CW skeleta are closed and cells form a disjoint partition Proposition
Euler characteristic is additive for finite CW pairs Proposition
Relative CW inclusions are cofibrations Proposition
Relative cellular homology computes relative singular homology Theorem
Relative homology of consecutive CW skeleta Theorem

Dependency tree · two levels

2 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

J. Peter May, A Concise Course in Algebraic Topology, Chapter 10 (standard reference, not scraped)