LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Compact generation preserves the cylinder and closed pushouts

Statement

Kification preserves compact Hausdorff test maps and cubical relative homotopy classes. For CGWH $X$ , the ordinary cylinder $X \times I$ is CGWH. If $A ↪ X$ is a closed inclusion of CGWH spaces and $f : A \to Y$ is continuous with $Y$ CGWH, the ordinary pushout $P = X \cup_{A} Y$ is CGWH; $Y \to P$ is a closed embedding and the pushout square is a pullback. These assertions apply to the cylinder attachments and closed-track, cone and suspension quotients below.

Facts & Assumptions

[F1]

Kification preserves compact tests, cylinders are CG, and closed inclusions remain closed under k-products. Kification, compact tests, and finite constructions

[F2]

CGWH quotients are characterized by closed fibre relations, and WH is characterized by a closed k-diagonal. Weak Hausdorff diagonals and closed quotients

[F3]

Products preserve fibrewise quotient maps in CG. Compact-test exponential law and products of quotient maps

[F4]

Quotient descent gives continuous factorizations and composites of quotients. For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map

Proof

Given: The spaces, maps, and hypotheses in the statement above.

1.1F1F2

The compact-test and relative-homotopy assertions are F1. Its ordinary cylinder is CG and agrees with the k-product; F2 gives WH of that product. Thus it is CGWH.

1.2F1

Put $S = X ⨿ Y$ , $P = (X ∖ A) ⨿ Y$ as a set, and let $q : S \to P$ be identity off $A$ and equal to $f$ on $A$ . Give $P$ the quotient topology. In the four clopen pieces of $S \times_{k} S$ , its equivalence relation is respectively $Δ_{X} \cup E_{f}$ , $G_{f}$ , $G_{f}^{o p}$ , and $Δ_{Y}$ , where $E_{f} = {(a, a^{'}) : f (a) = f (a^{'})}$ and $G_{f} = {(a, y) : f (a) = y}$ . These formulas include every fibre: only points of $A$ are identified with points of $Y$ , and two such points are equivalent exactly when their f-values agree.

2.1F1F2F4step 1.2

The sets $E_{f}$ and $G_{f}$ are inverse images of $Δ_{Y}$ in $A \times_{k} A$ and $A \times_{k} Y$ . They are closed there by F2, hence in the corresponding products with $X$ by F1. The two diagonals are closed. The relation in step 1.2 is therefore closed, and F2 proves $P$ CGWH. Maps from $S$ constant on that relation are precisely compatible maps from $X, Y$ , so F4 proves the pushout property in Top and in CGWH.

3.1F4step 1.2step 2.1

The map $j : Y \to P$ is injective. For closed $F \subseteq Y$ , one has $q^{- 1} (j (F)) = f^{- 1} (F) ⨿ F$ , which is closed in $S$ since $A$ is closed. Hence $j (F)$ is closed in $P$ . This proves that $j$ is a closed embedding. Set-theoretically $X \times_{P} Y$ consists exactly of $(a, f (a))$ for $a \in A$ . A continuous compatible pair from any space has X-component landing in the ordinary subspace $A$ and hence factors continuously there. This proves the pullback property. A closed $D \subseteq X$ disjoint from $A$ similarly embeds as a closed subspace, since it and each of its closed subsets are saturated for q.

4.1F1F2F3step 1.1step 2.1step 3.1∎

The attaching subspace $X \times {0}$ is closed, as are a WH basepoint track and finite unions of such tracks with a cone end. Collapsing any such closed subspace is the preceding pushout with a point. Thus the resulting cylinder, reduced-cylinder, cone and suspension quotients are CGWH. Their quotient products use F3 and retain the fibre coordinate; no whole product subspace is inadvertently collapsed. When $A = \emptyset$ , the formula is simply the disjoint union, and when $A = X$ , it is $Y$ .

Compact generation preserves the cylinder and closed pushouts

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources