The skeletal telescope projects by a homotopy equivalence of pairs

In $X\times [0,\mathrm{\infty })$ put $Y_i=T_X\cup X\times [i,\mathrm{\infty })$ and $B_i=X^i\times [i,i+1]\cup X\times \{i+1\}$. Then $Y_0=X\times [0,\mathrm{\infty })$ and ${\bigcap }_i{Y}_i=T_X$. We construct a slab strong deformation retraction onto $B_i$ that preserves every skeleton and $A$. Extending such a retraction by the identity on the rest of $Y_i$ gives $Y_i\to Y_{i+1}$: its intersection with the rest is contained in $B_i$.

Here is the controlled prism construction underlying the CW homotopy extension property. Rescale the slab coordinate to $u\in [0,1]$. On $D^n\times [0,1]$ project radially from $(0,-1)$ onto $E_n=\mathrm{\partial }{D}^n\times [0,1]\cup D^n\times \{1\}$. Explicitly put $\lambda (x,u)=2\mathrm{/}\max (2\mathrm{\parallel }x\mathrm{\parallel },u+1)$ and $r(x,u)=(\lambda x,-1+\lambda (u+1))$. We have $1\le \lambda \le 2$; the image lies on the side or top and $r$ fixes $E_n$. The straight-line homotopy $r_s=(1-s)\mathrm{i}\mathrm{d}+sr$ stays in the convex prism and fixes $E_n$. For a fixed $n>i$, applying this to all $n$-cell prisms gives a strong deformation retraction of $X^n\times [i,i+1]$ onto $X^{n-1}\times [i,i+1]\cup X^n\times \{i+1\}$. It glues along characteristic boundaries because the entire lower skeleton is fixed during this particular collapse. It also preserves $A$: an $n$-cell of $A$ and its attaching boundary both map into $A$.

For the fixed slab index $i$, perform the dimension-$n$ collapse during $[2^{-(n-i)},2^{-(n-i-1)}]$, for $n>i$, so higher dimensions collapse before lower ones. This specifies a homotopy on each $X^d\times [i,i+1]$: start with the identity until time $2^{-(d-i)}$ when $d>i$, then perform the finitely many collapses $n=d,d-1,\dots ,i+1$, always fixing the top; for $d\le i$ use the identity throughout. These homotopies agree on lower skeleta because a higher-dimensional collapse fixes its entire lower skeleton. Their endpoints lie in $B_i$ and they fix $B_i$ at every time. They preserve every skeleton and $A$, and assemble continuously: the restriction to every characteristic disk prism times the homotopy interval is a finite continuous concatenation. Products of a CW complex with the locally finite interval cell structures have their CW weak topology, so these restrictions test continuity, including at time zero. Thus this is the required single slab retraction.

Perform the retraction $Y_i\to Y_{i+1}$ during $[1-2^{-i},1-2^{-(i+1)}]$. A point initially in $X^d\times [0,\mathrm{\infty })$ stays in that skeleton and, by the end of stage $d$, lies in $Y_{d+1}\cap (X^d\times [0,\mathrm{\infty }))\subset T_X$. It is fixed thereafter. On each closed cell prism the infinite concatenation is therefore eventually stationary uniformly in its points. Define the time-one value by that stationary value. On each characteristic disk prism times the time interval the homotopy is a finite concatenation followed by a constant homotopy; the same CW product weak topology proves continuity at time one as well.

The whole homotopy fixes $T_X$ and preserves $A\times [0,\mathrm{\infty })$, ending there in $T_A$. It is a strong deformation retraction of pairs $(X\times [0,\mathrm{\infty }),A\times [0,\mathrm{\infty }))$ onto $(T_X,T_A)$. The ambient projection is a pair homotopy equivalence, with section at height zero and homotopy $(x,t,s)\mapsto (x,(1-s)t)$. Its composite with the telescope inclusion is the specified projection $p$, hence $p$ is a pair homotopy equivalence. Empty $X$, empty $A$, $A=X$, and zero-dimensional complexes are included by the same construction.