open ball neighborhood in CW complex

Question

In Hatcher's proof that CW complex is locally contractible (proposition A.4), he implicitly uses the fact that given a point $x \in X$, $X$ CW complex, $U$ an open neighborhood of $x$, there is an open ball of dimension $n$ called $N^n_\epsilon(x) \subset U$ for every $n$. Why is this true? I can see that this is true if $x$ lies in an $n$-cell, but for dimension $n+1,n+2,...$ I do not see why.

Answer 1

It's not implicit at all. He says exactly how the sets $N^n_\epsilon(x)$ are defined, half a page or so before the start of the proof of Proposition A.4.