False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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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.
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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 · two levels

11 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

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