Definition:Isolated Point (Complex Analysis)

Definition
Let $S \subseteq \C$ be a subset of the set of real numbers.

Let $z \in S$.

Then $z$ is an isolated point of $S$ iff there exists a neighborhood of $z$ in $\C$ which contains no points of $S$ except $z$:
 * $\exists \epsilon \in \R, \epsilon > 0: N_\epsilon \left({z}\right) \cap S = \left\{{z}\right\}$

By definition of neighborhood in the context of the complex plane under the usual (Euclidean) metric:
 * $N_\epsilon \left({z}\right) := \left\{{y \in S: \left \vert z - y \right \vert < \epsilon}\right\}$

it can be seen that this definition is compatible with that for a metric space:
 * $\exists \epsilon \in \R, \epsilon > 0: \left\{{y \in S: \left \vert z - y \right \vert < \epsilon}\right\} \cap S = \left\{{z}\right\}$