Given an absolute CW complex $X$ with skeleta $X_n$, I want to prove that I have the following homeomorphism: If $X_n$ arrises from $X_{n-1}$ by attaching $n$-cells with index set $J_n$, and with characteristic map $\chi_i$ for each $i\in J_n$, then I want to show that the map induced by the characteristic maps
$$
(J_n\times D^n)/(J_n\times\partial D^n)\to X_n/X_{n-1}
$$
is a homeomorphism. I've shown that this map is bijective, and continuity is easy too. However, when $J_n$ is infinite, I don't know how to argue that closed sets are mapped to closed sets (using that the induced characteri…