ExampleConstruction: Literature-sourcedVerification: 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.

A pullback in Top is the fibre product with the subspace topology inherited from the product

Example

For continuous maps $X \overset{f}{\to} Z \overset{g}{\leftarrow} Y$ , the pullback in $T o p$ is

$P = {(x, y) \in X \times Y : f (x) = g (y)}$

with the subspace topology inherited from the product topology on $X \times Y$ .

Facts & Assumptions

Given: The displayed continuous maps.

[F1]

A pullback represents compatible pairs of maps (Pullbacks and pushouts as limits and colimits of cospans and spans).

[L1]

Top-limits have the Set-limit as underlying set (Top is complete and cocomplete, and its underlying-set functor preserves all small limits and colimits).

[F2]

For $P = \prod_{i} X_{i}$ with the product topology of The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, a function $h : Z \to P$ is continuous exactly when every component $π_{i} \circ h$ is continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, claim 2).

[F3]

A map into a subspace is continuous exactly when its composite with the inclusion is continuous, provided its set map lands there (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

Verification

technique · universal property

1.1

The restricted coordinate projections $P \to X, Y$ are continuous by [F2] and [F3], and their composites with $f, g$ agree by definition of $P$ .

F2F3

1.2

If $r : T \to X$ and $s : T \to Y$ are continuous with $f r = g s$ , their unique Set-theoretic pairing lands in $P$ by [L1]. Its composite with $P ↪ X \times Y$ is $(r, s)$ , continuous by [F2], so [F3] makes the factor $T \to P$ continuous.

L1F2F3

2.1

Its uniqueness follows from uniqueness of its two coordinate functions. Thus [F1] identifies the displayed space as the pullback.

F1step 1.1step 1.2∎

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: 54 results over 11 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)