\begin{definition}[Definition:Deleted Neighborhood/Real Analysis]
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.
\end{definition}