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 $S$ if and only if there exists a neighborhood of $z$ in $\C$ which contains no points of $S$ except $z$:

$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \cap S = \set z$

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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:

$\exists \epsilon \in \R_{>0}: \set {y \in S: \cmod {z - y} < \epsilon} \cap S = \set z$