# Definition:Neighborhood (Complex Analysis)

*This page is about neighborhoods in the complex plane. For other uses, see Definition:Neighborhood.*

## Definition

Let $z_0 \in \C$ be a complex number.

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

The **$\epsilon$-neighborhood** of $z_0$ is defined as:

- $\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$

## Also known as

A **neighborhood** in this context is often referred to as an **open disk** (UK spelling: **open disc**).

Some sources introduce this concept as **$\delta$-neighborhood** (that is: **delta**), but it is the same thing.

## Also see

- Complex Plane is Metric Space: this definition is compatible with that of an open $\epsilon$-ball neighborhood in a metric space.

## Linguistic Note

The UK English spelling of **neighborhood** is **neighbourhood**.

## Sources

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