\begin{definition}[Definition:Limit Point/Complex Analysis]
Let $S \subseteq \C$ be a subset of the set of complex numbers.
Let $z_0 \in \C$. 
Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$.
Then $z_0$ is a '''limit point of $S$''' {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$:
:$\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$
that is:
:$\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$
Note that $z_0$ does not itself have to be an element of $S$ to be a '''limit point''', although it may well be.
Informally, there are points in $S$ which are arbitrarily close to it.
\end{definition}