\begin{definition}[Definition:Deleted Neighborhood/Complex Analysis]
Let $z_0 \in \C$ be a point in the complex plane.
Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$.
Then the '''deleted $\epsilon$-neighborhood''' of $z_0$ is defined as $\map {N_\epsilon} {z_0} \setminus \set {z_0}$.
That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself ''removed''.
It can also be defined as:
:$\map {N_\epsilon} {z_0} \setminus \set {z_0} : = \set {z \in A: 0 < \cmod {z_0 - z} < \epsilon}$
from the definition of $\epsilon$-neighborhood.
\end{definition}