Closure of an open cell in a CW complex

Question

Given a CW complex $X$ and any of its cells $e$, the closure $\bar{e}$ in $X$ is covered by finitely many open cells by "C". Can we prove that $\bar{e}$ is exactly a union of open cells, or, $(\bar{e}-e)$ a union of open cells of strictly lower dimensions? If not, is there any counterexample?

Answer 1

In general $\overline e$ won't be a union of open cells.

Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$ and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$ to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that isn't $e_0$. Then $\overline{e_2}=e_2\cup\{P\}$ and that isn't a union of open cells.