Compact generation preserves the cylinder and closed pushouts

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.

Put $S=X⨿Y$, $P=(X\setminus A)⨿Y$ 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 ${\mathrm{\Delta }}_X\cup E_f$, $G_f$, $G_f^{\mathrm{o}\mathrm{p}}$, and ${\mathrm{\Delta }}_Y$, where $E_f=\{(a,a^{\mathrm{\prime }}):f(a)=f(a^{\mathrm{\prime }})\}$ 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.

The sets $E_f$ and $G_f$ are inverse images of ${\mathrm{\Delta }}_Y$ 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.

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.

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 $A=\mathrm{\varnothing }$, the formula is simply the disjoint union, and when $A=X$, it is $Y$.