Definition:Convergent Sequence/Topology/Definition 1

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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 $\alpha \in S$ if and only if:

$\forall U \in \tau: \alpha \in U \implies \left({\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}\right)$


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