Definition:Isolated Point (Metric Space)/Subset

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Let $M = \struct {A, d}$ be a metric space.

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 $a$:

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

That is:

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

Also see

  • Results about isolated points can be found here.