Definition:Limit Point/Topology

Complex Analysis
Let $$S \subseteq \mathbb{C}$$ be a subset of the set of complex numbers.

Let $$z_0 \in \mathbb{C}$$.

Let $$N_{\epsilon} \left({z_0}\right)$$ be the $\epsilon$-neighborhood of $$z_0$$ for a given $$\epsilon \in \mathbb{R}$$ 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.