# Definition:Deleted Neighborhood/Complex Analysis

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## Definition

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.

## Also known as

A **deleted neighborhood** is also called a **punctured neighborhood**.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $1.$