In the proof of the cellular approximation theorem, Hatcher use the facts that if $Y$ is a CW-complex and $\ast$ is a point in an open $k$-cell $e^k$, then $Y-\ast$ deformation retracts into $Y-\overline{e^k}$, but he doesn't go into details. Clearly, the idea must be using the following retraction: $$r:D^k-\left\{\Phi^{-1}(\ast)\right\}\to \partial D^k$$ Where $\Phi^{-1}$ is the inverse of the restriction of the characteristic map to the open disk. I would like to define the retraction:
$$\rho:Y-\ast \to Y-\overline{e^k}, \ \ \ y\mapsto \begin{cases}y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if $\ \ y\in Y-e^k$} \\ \Phi(r(\Phi^{-1}(y))) \ \ \ \text{if $\ \ y\in e^k-\ast$} \end{cases}$$
But, I don't know how to prove that this map is continous. I can't apply gluing lemma because $Y-e^k$ is closed and $e^k-\ast$ is open in $Y-\ast$. Moreover $Y-\ast$ is not a CW-complex in a canonical way (at least not in a way that I know).