$$\phi(B^n)=\phi(\overline{B^n-S^{n-1}})=\overline{\phi(B^n-S^{n-1})}$$

where $\phi(B^n-S^{n-1})=c^n$

Prove that if $(X,\mathcal{C})$ is a CW complex, then the zero skeleton $X^0$ is discrete

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