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:


 * $N_\epsilon \left({z_0}\right) := \left\{{z \in \C: \left|{z - z_0}\right| < \epsilon}\right\}$

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 this is neighbourhood.