Definition:Convergent Sequence (Topology)

Definition

Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $\left \langle {x_k} \right \rangle$ be a sequence in $T$.

Then $\left \langle {x_k} \right \rangle$ converges to the limit $\alpha \in T$ if:

for any open set $U \in \vartheta$ such that $\alpha \in U$: $\exists N \in \R: n > N \implies x_n \in U$

This can be alternatively stated:

$\left \langle {x_k} \right \rangle$ converges to the limit $\alpha \in T$ if:

every open set in $T$ containing $\alpha$ contains all but a finite number of terms of $\left \langle {x_n} \right \rangle$.

Such a sequence is convergent.

Sources

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