$\epsilon$-Neighborhood

Definition

Given a real number $a \in \mathbb{R}$ and a positive number $\epsilon > 0$, the set

$$ V_{\epsilon}(a) = \{ x \in \mathbb{R} : |x - a| < \epsilon \} $$

is called the $\epsilon$-neighborhood of $a$.

Notice that $V_{\epsilon}(a)$ consists of all of those points whose distance from $a$ is less than $\epsilon$. Said another way, $V_{\epsilon}(a)$ is an interval, centered at $a$, with radius $\epsilon$.