Definition – Deleted neighborhood.

Definition

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

$N^*(x; \epsilon) = \{y \in \mathbb{R}: 0 < |x - y| < \epsilon\}$

Where $\epsilon > 0$ is a real number.

Notes

This definition is analogous to the definition of a $\epsilon$ -Neighborhood of $x$ , except that we do not include the point $x$ itself. This is indicated by the fact that $|x - y| < 0$ . Since $|x - y| \neq 0$ (because we haven’t written $\geq$ ), it follows that $y \neq x$ .