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.

CW complex with closure finiteness and weak topology

Definition

A CW complex is a Hausdorff space $X$ with a filtration $\emptyset = X^{- 1} \subseteq X^{0} \subseteq X^{1} \subseteq \dots, X = ⋃_{n \geq 0} X^{n},$ where $X^{0}$ is discrete and $X^{n}$ is obtained from $X^{n - 1}$ by attaching a (possibly infinite) family of $n$ -disks along maps $S^{n - 1} \to X^{n - 1}$ as in Cell attachment by a characteristic map. The images of the disk interiors are its open $n$ -cells $e_{α}^{n}$ , and the disk maps are their characteristic maps. It moreover satisfies: (C) each closed cell $\overline{e_{α}^{n}}$ meets only finitely many cells; and (W) $A \subseteq X$ is closed exactly when $A \cap \overline{e}$ is closed in $\overline{e}$ for every cell $e$ . These are closure finiteness and the weak topology.

Depends on

Cell attachment by a characteristic map

Used by

A cell decomposition without the weak topology need not be a CW complex Counterexample
Skeleta, CW subcomplexes, and relative CW complexes Definition
A compact subspace of a CW complex meets only finitely many cells Lemma
A CW complex is the colimit of its skeleta in the weak topology Proposition

Dependency tree · one level

1 result within one dependency step 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

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