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 $x \in S$.

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 $\sequence {x_n}$.

Also see

• 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