Definition:Convergent Sequence/Topology
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
page proving equivalence; the Archimedean principle is needed
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