\begin{definition}[Definition:Isolated Point (Complex Analysis)]
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 $z$:
:$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \cap S = \set z$
{{refactor|Extract the below into its own page|level = basic}}
By definition of neighborhood in the context of the complex plane under the usual (Euclidean) metric:
:$\map {N_\epsilon} z := \set {y \in S: \cmod {z - y} < \epsilon}$
it can be seen that this definition is compatible with that for a metric space:
:$\exists \epsilon \in \R_{>0}: \set {y \in S: \cmod {z - y} < \epsilon} \cap S = \set z$
Category:Definitions/Complex Analysis
\end{definition}