Convergent sequence - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A sequence $(x_n)$ in a topological space $X$ converges to $x\in X$ if, for every neighborhood $U$ of $x$ , there is $N\in\mathbb N$ such that $x_n\in U$ for all $n\geq N$ .

Metric spaces and uniqueness

In a metric space $(X,d)$ , this is equivalent to $d(x_n,x)\to 0$ . In a Hausdorff space, limits of convergent sequences are unique.

Examples

In $\mathbb R$ with the usual metric, the sequence $x_n=1/n$ converges to $0$ .
In a space with the discrete metric, a sequence converges if and only if it is eventually constant.