Neighborhood

Definition

Let $a\in \mathbb{R}$ and $\epsilon >0$. Then the $\epsilon$-neighborhood of $a$ is the set $V_{\epsilon }(a):=\{x\in \mathbb{R}:\mid x-a|<\epsilon \}$

Remark

This essentially means that $x$ is “close” to $a$. $V_{\epsilon }(a)$ is the set of all real number within $\epsilon$ unit “distance” of $a$.

For example 2-neighborhood of 3 means numbers within 2 unit distance of 3, which ranges from 1 to 5.