False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: the underlying-set functor Top→Set fails to preserve some small limit

Statement refuted

The underlying-set functor $U : T o p \to S e t$ does not preserve all small limits.

Facts & Assumptions

Given: The underlying-set functor $U$ .

[L1]

$T o p$ is complete, and $U$ preserves every small limit and colimit (Top is complete and cocomplete, and its underlying-set functor preserves all small limits and colimits).

[F1]

Preservation means the image of every limiting cone is limiting (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

Refutation

technique · apply the proved construction

1.1

For each small Top-diagram, [L1] constructs its limit on exactly the underlying Set-limit and adds the initial topology. Applying $U$ removes that topology and leaves the Set-limit cone unchanged.

2.1

Therefore every image cone is limiting in $S e t$ , which is preservation by [F1]. The asserted counterexample cannot exist, so the statement is false.

F1step 1.1

3.1

This does not claim reflection: a Set-limiting underlying cone need not already carry the initial topology required for a Top-limit.

F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, Proposition 3.6.2 (standard reference, not scraped)