I was reading Hatcher and saw his proof that CW complex is locally contractible. I did not quite get it, so I decided to prove it myself. Here is my attempt:
Pick any $x \in X$ with $X$ is a CW complex. Then $x$ is contained in some $n$-cell $e^n_\alpha$, so I can find an open ball $B^n(x)$ around $x$ that is completely contained inside $e^n_\alpha \subset \overline{e^n_\alpha}$. Define $E^n = B^n(x)$.
$x$ must be disjoint from all other $n$-cell and any cells of dimension less than $n$, and $x$ might lie in the boundary of cells of dimension greater than $n$. For each of these cells $e^k_\alpha$ of dimension $k > n$, I find a ball $B^k_\alpha(x)$ with radius less than 1. I then define $E^{k} = E^{k-1} \bigcup\limits_{\alpha} (B^k_\alpha(x) \bigcap \overline{e^k_\alpha})$.
Setting $E = \bigcup E^k$, because $E$ intersects with any cell is $B^k_\alpha(x) \bigcap \overline{e^k_\alpha}$, then $E$ is open. Finally, since each piece of $E$ in each cell is a restriction of a ball $B^k_\alpha(x)$ to $\overline{e^k_\alpha}$ and $B^k_\alpha(x)$ deformation retracts to x, then E deformation retracts to x. Is this correct?