Definition:Accumulation Point

Definition

Let $\struct {S, \tau}$ 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 $x \in S$.

Then $x \in S$ is an accumulation point of $\sequence {x_n}$ if and only if:

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

Accumulation Point of Set

Let $x \in S$.

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

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

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

Also known as

An accumulation point is also known as a cluster point, but that term has a number of different meanings.

Some sources refer to an accumulation point as a limit point, but $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to maintain a distinction between the two concepts.

Also see

• Definition:Omega-Accumulation Point

• Results about accumulation points can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): accumulation point