Let $A$ be a subset of $X^{0}$ and $c$ be a cell of $\mathcal{C}$. For each $c$, let $X_c\subseteq X^{0}$ be a finite subcomplex contains $\bar{c}$. Note that $A\cap X_c$ is a finite union of cells. As $X$ is a Hausdorff space and finite sets in Hausdorff space is closed, $A\cap X_c$ is closed. For every cell $c_i\in A\cap X_c$, we have $\bar{c}_i\in A\cap X_c$ for $i=1,\dots,m$.
Moreover, we have $A\cap\bar{c}=(A\cap X_c)\cap\bar{c}$ is closed but also finite because it is a subset of $\bar{c}$. So, the set $A$ is closed because $X$ has the weak topology determines by $X_c$. Therefore, the zero skeleton $X^{0}$ is a discrete closed subset of $X$.
