Definition:Limit Point/Metric Space

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

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.

Then $\alpha$ is a limit point of $A$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$:

$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap A \ne \O$

that is:

$\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$

Note that $\alpha$ does not have to be an element of $A$ to be a limit point.

(Informally speaking, $\alpha$ is a limit point of $A$ if there are points in $A$ that are different from $\alpha$ but arbitrarily close to it.)