Definition:Deleted Neighborhood (Real Analysis)

Definition

Let $\alpha \in \R$ be a real number.

Let $N_\epsilon \left({\alpha}\right)$ be the $\epsilon$-neighborhood of $\alpha$:

$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon .. \alpha + \epsilon}\right)$

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

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

It can also be defined as:

$N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left\{{x \in \R: 0 < \left \vert{\alpha - x}\right \vert < \epsilon}\right\}$

or

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

from the definition of $\epsilon$-neighborhood.