PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

A CW complex is the colimit of its skeleta in the weak topology

Statement

A map $f : X \to Y$ from a CW complex is continuous if and only if every restriction $f ∣_{X^{n}} : X^{n} \to Y$ is continuous.

Facts & Assumptions

Given: A function $f : X \to Y$ whose skeletal restrictions are continuous.

Proof

technique · direct

1.1

Necessity follows by restriction. Conversely, $f ∣_{\overline{e}}$ is continuous because $\overline{e}$ lies in the skeleton of its dimension.

given

2.1

For closed $C \subseteq Y$ , each $f^{- 1} (C) \cap \overline{e}$ is closed in $\overline{e}$ by step 1.1; the weak-topology clause in CW complex with closure finiteness and weak topology makes $f^{- 1} (C)$ closed.

step 1.1∎

Depends on

Used by

A cell decomposition without the weak topology need not be a CW complex Counterexample

Dependency tree · two levels

3 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)