Definition:Isolated Point (Metric Space)/Space

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Definition

Let $M = \left({A, d}\right)$ 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 $a$:

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

That is:

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


Also see

  • Results about isolated points can be found here.


Sources