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