Definition:Limit Point/Metric Space/Definition 1

Definition

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

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

Let $\alpha \in S$.

$\alpha$ 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 $A$:

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

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.)

Also see

• Equivalence of Definitions of Limit Point in Metric Space

• Results about limit points in metric spaces 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: Definition $6.6$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.9$