Definition:Isolated Point (Complex Analysis)

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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:

$\exists \epsilon \in \R, \epsilon > 0: \left\{{y \in S: \left \vert z - y \right \vert < \epsilon}\right\} \cap S = \left\{{z}\right\}$