Definition:Isolated Point (Metric Space)/Subset
< Definition:Isolated Point (Metric Space)(Redirected from Definition:Isolated Point of Subset of Metric Space)
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Definition
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.