# Definition:Accumulation Point

## Definition

Let $\paren { S, \tau }$ be a topological space.

Let $A \subseteq S$.

### Accumulation Point of Sequence

Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.

Let $x \in S$.

Suppose that:

$\forall U \in \tau: x \in U \implies \set {n \in \N: x_n \in U}$ is infinite

Then $x$ is an accumulation point of $\sequence {x_n}$.

### Accumulation Point of Set

Let $x \in S$.

Then $x$ is an accumulation point of $A$ if and only if:

$x \in \cl {A \setminus \set x}$

where $\operatorname{cl}$ denotes the (topological) closure of a set.

## Also see

• Results about Accumulation Points can be found here.