Definition:Convergent Sequence/Topology

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Definition

Definition 1

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:

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


Definition 2

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.


Such a sequence is convergent.


Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.


Also see