Definition:Isolated Point (Metric Space)/Space
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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.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets