If $(X,\mathcal{C})$ is a CW complex, then $X^{(0)}$ is a discrete closed subset of $X$.
Let $A\subset X^{0}$. For any cell $c\in\mathcal{C}$, let $X_c$ be a finite subcomplex that contains $\bar{c}$. Note that $A\cap X_c=X_c\setminus(X_c\cap A)$. As $X_c$ is finite, we have $X_c\setminus(X_c\cap A)$ is finite. Additionally, $X$ is Hausdorff, $A\cap X_c=X_c\setminus(X_c\cap A)$ is closed because finite set is closed in Hausdorff space. Thus, $A$ is closed by the weak topology. Moreover, $A\cap\bar{c}=A\cap(X_c\cap\bar{c})$ which is finite.
this is what I have so far, and I don't know how to move forward.
Edit one: changed discrete finite to discrete closed.
Edit two: found an error in the proof I have, $A\cap X_c$ doesn't eqaul to $X_c\setminus(A\cap X_c)$