Definition:Neighborhood (Complex Analysis)

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.

Linguistic Note

The UK English spelling of neighborhood is neighbourhood.