Let $X$ be a CW complex and $X^n$ the $n$-skeleton of $X$ for all $n$. Hatcher demonstrates that $X$ is locally contractible by constructing a deformation retract from a certain open neighborhood $N_\epsilon(x)$ of a point $x \in X$ to $x$. The neighborhood is constructed as the union $\bigcup_{n}N_\epsilon^n(x)$, where each $N_\epsilon^n(x)$ is a neighborhood of $x$ in $X^n$, and Hatcher first builds deformation retracts $F_n$ from $N_\epsilon^n(x)$ to $N_\epsilon^{n-1}(x)$. He then stitches these together to a deformation retract of $N_\epsilon(x)$ by performing $F_n$ during the $t$-interval $[1/2^n, 1/2^{n-1}]$ and leaving the points of $N_\epsilon^n(x) - N_\epsilon^{n-1}(x)$ stationary outside of this interval.
I'm having trouble verifying that this map is continuous given its weird behavior near $t = 0$. Deformation retracts constructed this way are not continuous in general, and so there's something special about the weak topology on $X$ that makes it work in this case - I just can't figure out what that is.