Definition:Isolated Point

Definition

Topology

$x \in S$ is an isolated point of $T$ if and only if:

$\exists U \in \tau: U = \set x$

That is, if and only if there exists an open set of $T$ containing no points of $S$ other than $x$.

Metric Space

$a \in A$ is an isolated point of $M$ if and only if there exists an open $\epsilon$-ball of $x$ containing no points other than $a$:

$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a = \set a$

That is:

$\exists \epsilon \in \R_{>0}: \set {x \in A: \map d {x, a} < \epsilon} = \set a$

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

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

Real Analysis

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

Let $\alpha \in S$.

Then $\alpha$ is an isolated point of $S$ if and only if there exists an open interval of $\R$ whose midpoint is $\alpha$ which contains no points of $S$ except $\alpha$:

$\exists \epsilon \in \R_{>0}: \openint {\alpha - \epsilon} {\alpha + \epsilon} \cap S = \set \alpha$