In a CW complex, what is the closure of a cell $e_{\alpha}^n$?

Question

Is the closure of a cell $e_{\alpha}^n$ exactly $\Phi_{\alpha}^n(D_{\alpha}^n)$?I can only show $\Phi_{\alpha}^n(D_{\alpha}^n)\subset\overline{e_{\alpha}^n}$.

Answer 1

Note that $D^n_\alpha$ is compact, so $\Phi^n_\alpha(D^n_\alpha)$ is compact and thus $\overline{e^n_\alpha}\subset\Phi^n_\alpha(D^n_\alpha).$ And this is sufficient, since $CW$ complexes are Hausdorff (see Proposition A.3 of Hatcher's book for the details).