Definition:Convergent Sequence/Topology/Definition 2

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Definition

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

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


Then $\left \langle {x_n} \right \rangle$ converges to the limit $\alpha \in S$ if and only if:

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


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