Definition:Deleted Neighborhood (Complex Analysis)

Definition

Let $x \in \C$ be a point in the complex plane.

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

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:

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

from the definition of $\epsilon$-neighborhood.