This is from the Numerical Optimization's third lecture at this point.
Here is a reprint.
Let $S$ be subset of $\mathbb{R}^n$. $x$ which is an element of $\mathbb{R}^n$ belongs to the closure of $S$, denoted $\operatorname{cl}(S)$, if for each $e > 0$, $S \cap B[x,e] \neq\varnothing$.
Example: Let $S = (1,2] \cup [3,4)$. Then $\operatorname{cl}(S) = [1,2] \cup [3,4]$
I am having difficulty understanding why this definition means what the examples show. Wouldn't the above hold true for essentially every $x$ as long as $e > 0$, and that would include points far away from the boundary of $S$?