Definition:Limit Point (Metric Space)

Definition

Let $M = \left({S, d}\right)$ be a metric space.

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

Let $\alpha \in S$.

Then $\alpha$ is a limit point of $A$ iff every deleted $\epsilon$-neighborhood $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$ of $\alpha$ contains a point in $A$ other than $\alpha$:

$\forall \epsilon \in \R, \epsilon > 0: N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} \cap A \ne \varnothing$

that is:

$\forall \epsilon \in \R, \epsilon > 0: \left\{{x \in A: 0 < d \left({x, \alpha}\right) < \epsilon}\right\} \ne \varnothing$

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.)