Let us refer to the latest online version of Hatcher's Algebraic Topology, and the proof of Proposition A.4, as well as the inductive construction of an $\epsilon$-neighborhood of a subset of a CW complex, right above Proposition A.3.
Let $X$ be a CW complex, $x$ a point of $X$ and $U$ an open neighborhood of $X$ that contains $x$. The goal is to show that for small enough $\epsilon$ we can take $N_{\epsilon}(x)$ to be a subset of $U$. Hatcher says that we can do this inductively by requiring that for each $n$ the closure of $N_{\epsilon}^n(x)$ is inside $U$. Why do we need this condition, i.e., why not just take $\epsilon$ small enough such that simply $N_{\epsilon}^n(x)$ is inside $U$?