Definition:Isolated Point

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Topology

Let $T = \left({S, \tau}\right)$ be a topological space.


Isolated Point of Subset

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

$\exists U \in \tau: U \cap H = \left\{{x}\right\}$

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


Isolated Point of Space

When $H = S$ the definition applies to the entire topological space $T = \left({S, \tau}\right)$:


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

$\exists U \in \tau: U = \left\{{x}\right\}$

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


Metric Space

Let $M = \left({A, d}\right)$ be a metric space.


Isolated Point in Subset

Let $S \subseteq A$ be a subset of $A$.

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

$\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) \cap S = \left\{{a}\right\}$

That is:

$\exists \epsilon \in \R_{>0}: \left\{{x \in S: d \left({x, a}\right) < \epsilon}\right\} = \left\{{a}\right\}$


Isolated Point in Space

When $S = A$ this reduces to:

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

$\exists \epsilon \in \R_{>0}: B_\epsilon \left({a}\right) = \left\{{a}\right\}$

That is:

$\exists \epsilon \in \R_{>0}: \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\} = \left\{{a}\right\}$


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 $z$:

$\exists \epsilon \in \R, \epsilon > 0: N_\epsilon \left({z}\right) \cap S = \left\{{z}\right\}$


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 $\alpha$:

$\exists \epsilon \in \R_{>0}: \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right) \cap S = \left\{{\alpha}\right\}$