# 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 $\map {N_\epsilon} {z_0}$ 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 $S$** if and only if *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.

## 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.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $2.$