Definition:Neighborhood (Complex Analysis)

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

Let $$\epsilon \in \R: \epsilon > 0$$ be a positive real number.

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


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

In this context, a neighborhood is often referred to as an open disk (UK spelling: open disc).

Linguistic Note
The UK English spelling of this is neighbourhood.