$epsilon.alt$-neighborhood

Definition

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

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

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

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