On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$:
A set $U$ in $\mathbb{R}^n$ will be open if given any $a \in \mathbb{R}^n$, there is a real number $\varepsilon > 0$ such that $\{x : |x-a| < \varepsilon\}$ is contained in $U$.
This looks similar to the other definitions that I've seen that give the open sets of $\mathbb{R}^n$ as the open balls of $\mathbb{R}^n$. The part that throws me is "if given any a...". The way the definition is worded, it looks like he is saying that given any open set $U$ and any point $a$ in $\mathbb{R}^n$, there is an open ball around $a$ contained in $U$. But that doesn't align with how open sets are usually defined, and it certainly doesn't look useful.
I guess my question comes down to this: am I misreading the definition, is the definition worded sloppily, or is something else going on here?