Definition:Deleted Neighborhood/Real Analysis

Definition
Let $x \in \R$ be a real number.

Let $N_\epsilon \left({x}\right)$ be the $\epsilon$-neighborhood of $x$:
 * $N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon .. \alpha + \epsilon}\right)$

Then the deleted $\epsilon$-neighborhood of $x$ is defined as $N_\epsilon \left({x}\right) \setminus \left\{{x}\right\}$.

That is, it is the $\epsilon$-neighborhood of $x$ with $x$ itself removed.

It can also be defined as:
 * $N_\epsilon \left({x}\right) \setminus \left\{{x}\right\} : = \left\{{y \in \R: 0 < \left \vert{x - y}\right \vert < \epsilon}\right\}$

or
 * $N_\epsilon \left({x}\right) \setminus \left\{{x}\right\} : = \left({\alpha - \epsilon .. \alpha}\right) \cup \left({\alpha .. \alpha + \epsilon}\right)$

from the definition of $\epsilon$-neighborhood.