Definition:Convergent Sequence/Topology/Definition 1

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

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

Then $\left \langle {x_n} \right \rangle$ converges to the limit $\alpha \in S$ iff:
 * $\forall U \in \tau: \alpha \in U \implies \left({\exists N \in \R: \forall n \in \N: n > N \implies x_n \in U}\right)$

Source

 * : $4.1$: Motivation: Definition $4.1.2$