Definition:Accumulation Point/Sequence

Definition

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

Let $A \subseteq S$.

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

Also known as

An accumulation point is also known as a cluster point.

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

• Accumulation Point of Sequence is not necessarily Limit Point

• Results about accumulation points can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): accumulation point
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): accumulation point
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): limit point (accumulation point, cluster point): 1. (of a sequence)