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 deleted $\epsilon$-neighborhood $N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\}$ of $z_0$ contains a point in $S$ other than $z_0$:
 * $\forall \epsilon \in \R, \epsilon > 0: \left({N_\epsilon \left({z_0}\right) \setminus \left\{{z_0}\right\}}\right) \cap S \ne \varnothing$

that is:
 * $\forall \epsilon \in \R, \epsilon > 0: \left\{{z \in S: 0 < \left|{z - z_0}\right| < \epsilon}\right\} \ne \varnothing$

Note that $z_0$ does not 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.

Also known as
A limit point is also known as a cluster point.

Some sources also use the term accumulation point for limit point, but as this has a slightly different definition in more general topology, it is recommended that this not be used in this context.