Definition:Deleted Neighborhood/Real Analysis

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

Let $\map {N_\epsilon} \alpha$ be the $\epsilon$-neighborhood of $\alpha$:
 * $\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$

Then the deleted $\epsilon$-neighborhood of $\alpha$ is defined as:
 * $\map {N_\epsilon} \alpha \setminus \set \alpha$.

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

It can also be defined as:
 * $\map {N_\epsilon} \alpha \setminus \set \alpha : = \set {x \in \R: 0 < \size {\alpha - x} < \epsilon}$

or
 * $\map {N_\epsilon} \alpha \setminus \set \alpha : = \openint {\alpha - \epsilon} \alpha \cup \openint \alpha {\alpha + \epsilon}$

from the definition of $\epsilon$-neighborhood.