Definition – Neighborhood. – Math-reference.net

Definition

A neighborhood of a real number $x$ is a set denoted by $N(x;\epsilon)$ , where

$\huge{N(x;\epsilon) = \{y \in \mathbb{R}: |x - y| < \epsilon\}}$

and $\Large{\epsilon} > 0$ is a real number. $\Large{\epsilon}$ is referred to as the radius of the neighborhood.

Notes

This definition tells us that a neighborhood is just a set of numbers within $\epsilon$ of $x$ . Do not get caught up on what the value of $\epsilon$ actually is; we do not generally define exactly how big the interval should be. Rather, we usually use this definition to prove other theorems without making reference as to a specific $\epsilon$ .

Also see deleted neighborhood.