ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13

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 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 · two levels

22 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)