Closed $n$ cell and Hausdorff property proof
Solution 1:
If $Y$ is not Hausdorff, then there is no reason to expect that $\overline{\Phi(D^n\backslash \mathbb{S}^{n-1}})=\Phi(\overline{D^n\backslash \mathbb{S}^{n-1}})$ in $Y$. It is only correct for the closure in the subspace $E$. Example:
Let $Y = D^n + S$, where $S$ is any non-empty set. Give $Y$ the topology consisting of all open subsets of $D^n$ and $Y$ itself.