Definition:Limit Point/Topology/Set/Definition 1

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

A point $x \in S$ is a limit point of $A$ every open neighborhood $U$ of $x$ satisfies:
 * $A \cap \paren {U \setminus \set x} \ne \O$

That is, every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.

More symbolically, a point $x \in S$ is a limit point of $A$

$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$

Also see

 * Equivalence of Definitions of Limit Point