Definition:Limit Point/Complex Analysis

Definition
Let $S \subseteq \C$ be a subset of the set of complex numbers.

Let $z_0 \in \C$.

Let $N_\epsilon \left({z_0}\right)$ 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 $N_\epsilon \left({z_0}\right)$ contains a point in $S$ other than $z_0$.

Note that $z_0$ does not have to be an element of $S$ to be a limit point, although it may well be.

The point is (no pun intended) that there are points in $S$ which are arbitrarily close to it.