This question has been answered here, but the method used is different than the one I learned in my algebraic topology course, which I believe I should use. I showed that given that $Y$ is a CW complex (satisfies characteristic map axiom, closure finiteness, and weak topology axioms), the characteristic map $f: D_\alpha^k\to Y$ lifts to a map $\tilde{f}: D_\alpha^k\to X$ which satisfies the characteristic map axiom, so the cells are $e = \tilde{f}(int(D^k_\alpha))$. Now I want to show that $A\subseteq X$ is closed iff $A\cap \bar{e} = A\cap \tilde{f}(D^k_\alpha)$ is closed, which is my definition of weak topology. The forward direction is simple, both $A$ and $\bar{e}$ are closed so their intersection is closed.
The reverse direction is giving me trouble. I want to use the fact that $p$ restricted to certain open sets gives a homeomorphism, and if I show $A\cap \bar{e}$ is in one of such sets, then it should be closed since $f(D^k_\alpha)$ is also closed (or something along those lines). Any help is appreciated, thank you!