Relative CW inclusions are cofibrations

First establish the needed quotient-cylinder mapping criterion. Let $q:E\to B$ be any quotient map and let $h:B\times I\to Y$ be a function such that $h\circ (q\times {\mathrm{i}\mathrm{d}}_I)$ is continuous. For each $b$, surjectivity supplies a representative $e$ and shows that the path $t\mapsto h(b,t)$ is continuous. No simultaneous selection of representatives is needed. The continuous transpose upstairs is $\widehat{h}\circ q:E\to P$. For every open $V\subseteq P$, its inverse image is $q^{-1}[{\widehat{h}}^{-1}[V]]$, so the quotient criterion makes ${\widehat{h}}^{-1}[V]$ open. Thus $\widehat{h}$ and then $h$ are continuous. No closedness hypothesis on $q$ is used.

For $n\ge 1$, put $T_n=(D^n\times \{0\})\cup (S^{n-1}\times I)$. For $u\in D^n$ and $t\in I$, define $a=\max \{1-t\mathrm{/}2,\mathrm{\parallel }u\mathrm{\parallel }\}$, $\lambda =1\mathrm{/}a$, and $r_n(u,t)=(\lambda u,2+\lambda (t-2))$. Since $1\mathrm{/}2\le a\le 1$, this formula is continuous and defined on the entire cylinder. Moreover $1\le \lambda \le 2\mathrm{/}(2-t)$, so its height lies in $[0,t]$, and $\lambda \mathrm{\parallel }u\mathrm{\parallel }\le 1$. If $a=1-t\mathrm{/}2$ the height is zero, while if $a=\mathrm{\parallel }u\mathrm{\parallel }$ the norm of the first coordinate is one; hence the image lies in $T_n$. If $t=0$ or $\mathrm{\parallel }u\mathrm{\parallel }=1$, then $a=1$ and the point is fixed. Thus $r_n$ is a retraction onto $T_n$. For $n=0$ set $r_0(\ast ,t)=(\ast ,0)$.

We specify compatible continuous maps $H_n:X^n\times I\to Y$, with $H_n(x,0)=g(x)$ and $H_n=H$ on $(A\cap X^n)\times I$. On the discrete zero skeleton prescribe $H_0(v,t)=H(v,t)$ for $v\in A$ and $H_0(v,t)=g(v)$ otherwise. Every vertex cylinder is open, so this is continuous.

We justify the infinite-stage continuity using the weak topology on $X$, not an assumed weak topology on $X\times I$. Each characteristic map $\varphi :D^n\to \bar{e}$ is a quotient map. Indeed it is a continuous surjection; if $F\subseteq D^n$ is closed, it is compact. Its image is compact by pulling any open cover of $\varphi [F]$ back to $F$ and retaining the images' finitely many covering members. This open-cover argument works for arbitrary targets, without a metrizability hypothesis on $\bar{e}$. Since $\bar{e}$ is a subspace of the Hausdorff space $X$, $\varphi [F]$ is closed there. Thus $\varphi$ is closed; if ${\varphi }^{-1}[C]$ is closed, then $C=\varphi [{\varphi }^{-1}[C]]$ is closed, proving the quotient criterion.

Given such $H_{n-1}$ for $n\ge 1$, let ${\varphi }_{\alpha }:D^n\to X^n$ be a supplied characteristic map and $f_{\alpha }:S^{n-1}\to X^{n-1}$ its boundary map. If its open cell belongs to $A$, use $K_{\alpha }(u,t)=H({\varphi }_{\alpha }(u),t)$; this is defined on the whole disk cylinder because the closed cell lies in $A$. For every other cell define $b_{\alpha }:T_n\to Y$ by $b_{\alpha }(u,0)=g({\varphi }_{\alpha }(u))$ and $b_{\alpha }(u,t)=H_{n-1}(f_{\alpha }(u),t)$ for $u\in S^{n-1}$. These two continuous prescriptions agree on the corner and their domains form a two-piece closed cover of $T_n$, so $b_{\alpha }$ is continuous. Set $K_{\alpha }=b_{\alpha }\circ r_n$.

The maps $H_{n-1}$ and all $K_{\alpha }$ agree on the attaching identifications, including for cells belonging to $A$, and so determine a function $H_n:X^n\times I\to Y$. The map from $(X^{n-1}\bigsqcup {\coprod }_{\alpha }{D}^n)\times I$ is continuous: each summand cylinder is open and carries the prescribed continuous map. The attachment map $X^{n-1}\bigsqcup {\coprod }_{\alpha }{D}^n\to X^n$ is quotient, so step 1.1 proves continuity of $H_n$. Its time-zero values are $g$, it extends $H_{n-1}$, and it equals $H$ on $(A\cap X^n)\times I$, since every cell of this intersection is either in the previous skeleton or among the cells on which $K_{\alpha }$ was prescribed from $H$.

These formulas define the successor map uniquely from the preceding stage; they do not choose one extension from a nonempty set of extensions. Apply ordinary recursion, recording the stage number together with the partial map, to obtain all $H_n$. The state space can be taken to be the set of valid stage-number/partial-map pairs, with the successor just constructed. Define $\tilde{H}(x,t)=H_n(x,t)$ whenever $x\in X^n$. Compatibility makes this independent of $n$, gives $\tilde{H}(x,0)=g(x)$, and gives $\tilde{H}{\mathrm{\mid }}_{A\times I}=H$.

Every path $t\mapsto \tilde{H}(x,t)$ is continuous by some $H_n$, so the transpose $G:X\to P$ is well defined. For each characteristic map $\varphi :D^n\to X$, the map $\tilde{H}\circ (\varphi \times {\mathrm{i}\mathrm{d}}_I)$ is continuous by construction, and [L3] makes $G\circ \varphi$ continuous. By the quotient property from step 1.4, $G{\mathrm{\mid }}_{\bar{e}}$ is continuous. Consequently, for every closed $C\subseteq P$, the set $G^{-1}[C]$ meets every closed cell in a relatively closed set. The CW weak topology implies $G^{-1}[C]$ is closed in $X$. Thus $G$ is continuous. This argument permits arbitrary families of cells and unbounded dimension.

The exponential law applied once more, now to $G:X\to P$, proves that $\tilde{H}:X\times I\to Y$ is continuous. Its restrictions established in step 4.1 solve every homotopy-extension problem in the statement, which is the asserted cofibration property.