\begin{definition}[Definition:Limit Point/Topology/Set/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
A point $x \in S$ is a '''limit point of $A$''' {{iff}} every open neighborhood $U$ of $x$ satisfies:
:$A \cap \paren {U \setminus \set x} \ne \O$
That is, {{iff}} 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 $A$''' {{iff}}
$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$
\end{definition}