CW-pairs are good pairs Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?

Let $(X,A)$ be the CW-pair. We can inductively construct an open neighborhood $N_\epsilon(A)$, where $\epsilon$ is function assigning to each cell $e_\alpha$ a positive $\epsilon_\alpha<1$.
Assume that $N^n_\epsilon(A)$ has been constructed, a neighborhood of $A\cap X^n$ in $X^n$, starting the process with $N^0_\epsilon(A)=A\cap X^0$. Then we define $N^{n+1}_\epsilon(A)$ by specifying its preimage under the characteristic map $\Phi_\alpha:D^{n+1}\to X$ of each cell $e_\alpha^{n+1}$. This will be a product $(1-\epsilon_\alpha,1]\times\Phi^{-1}_\alpha(N^{n}_\epsilon(A))$ with respect to 'spherical' coordinates $(r,\theta)$
in $D^{n+1}$, where $r\in[0,1]$ is the radial coordinate and $\theta$ lies in $\partial D^{n+1}=S^n$. Obviously, $\Phi^{-1}_\alpha(N^{n+1}_\epsilon(A))$ will be defined as all of $D^{n+1}$ if $e_\alpha$ is a cell in $A$.
We can perform the deformation retraction of $N_\epsilon^{n+1}(A)$ onto $N^n_\epsilon(A)$ during the time interval $[1/2^{n+1},\ 1/2^n]$. So this is a map $N^{n+1}_\epsilon(A))\times[1/2^{n+1},\ 1/2^n]\to N_\epsilon^{n+1}(A)$, call it $h^n$ between the identity on $N^{n+1}_\epsilon$ and a retraction $r^n$ of $N^{n+1}_\epsilon$ to $N^n_\epsilon$ at the time $1/2^n$. Since the next homotopy $h^{n-1}$ is defined as a map $N^n_\epsilon\times[1/2^n,\ 1/2^{n-1}]\to N^n_\epsilon$, but we would like to have a map $N^{n+1}_\epsilon(A))\times[1/2^{n+1},\ 1/2^{n-1}]\to N_\epsilon^{n+1}(A)$, we can simply compose $h^{n-1}$ with $r^n$ for $t\ge/2^n$. In the end, since we actually want a map $h:N_\epsilon(A)\times I\to N_\epsilon(A)$, all we have to do is to compose $h^{n-1}(-,t)\circ r^n\circ...\circ r^m$ for a point $x\in N^{m+1}(A)$ in order to see where it gets mapped to at $t\in[1/2^n,\ 1/2^{n-1}]$. This is continuous since its composition with each characteristic map is continuous and CW-complexes have the final topology.
