# Definition:Isolated Point (Metric Space)/Space

## Definition

Let $M = \struct {A, d}$ be a 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:

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

## Also see

• Results about isolated points can be found here.