Definition:Accumulation Point of Sequence

Definition

Let $\left({S, \tau}\right)$ be a topological space.

Let $A \subseteq S$.

Let $\left \langle {x_n} \right \rangle_{n \mathop \in \N}$ be an infinite sequence in $A$.

Let $\alpha \in A$.

Suppose that:

$\forall U \in \tau: \alpha \in U \implies \left\{{n \in \N: x_n \in U}\right\}$ is infinite

Then $\alpha$ is an accumulation point of $\left \langle {x_n} \right \rangle$.

Also see

• Results about accumulation points can be found here.

Sources

• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 1$: Limit Points