In Kochman's stable homotopy theory, pg 121 prop 4.24
We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge -1} [n-1,n]\ltimes X^n \subseteq [-2,\infty) \ltimes X$$ where we let $X^{-1}=*$.
My confusion. Today I have learnt that $\ltimes$ denotes the half smash product: $$(X,x) \ltimes Y := X \wedge Y_+ \cong X \times Y/\{ x\} \times Y$$
Now how does $[n-1,n] \ltimes X^n$ live in $[-2,\infty) \ltimes X$?