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?
 A: 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.
A: Have you checked the appendix? I think there are more explanations there about CW complexes topology.
A: This should have been a comment to Stefan Hamcke's answer.
The homotopy in Stefan's answer seems to be identical to the one given in Hatcher's appendix.  Several details are left to the reader, though.  In particular, Hatcher does not explain why his homotopy is continuous.
Stefan Hamcke suggests deducing this from universal properties, but the topology on $I \times N_\epsilon(A)$ is obtained by combining three constructions: product topology, subspace topology, and quotient topology.  These don't always commute (and shouldn't be expected to, since it's commuting a limit and a colimit!)
Universal properties gives continuity when $I \times N_\epsilon(A)$ is given the quotient topology of the subspace topology, whereas what we need is continuity in the subspace topology of the quotient topology.
