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Limit of a sequence
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==Topological spaces== ===Definition=== A point <math>x \in X</math> of the topological space <math>(X, \tau)</math> is a '''{{visible anchor|Limit of a sequence in a topological space|text=limit}}''' or '''{{visible anchor|Limit point of a sequence|text=limit point}}'''{{sfn|Dugundji|1966|pp=209-210}}{{sfn|Császár|1978|p=61}} of the [[sequence]] <math>\left(x_n\right)_{n \in \N}</math> if: :For every [[Topological neighbourhood|neighbourhood]] <math>U</math> of <math>x</math>, there exists some <math>N \in \N</math> such that for every <math>n \geq N</math>, we have <math>x_n \in U</math>.<ref>{{cite book|last1=Zeidler|first1=Eberhard|title=Applied functional analysis : main principles and their applications|date=1995|publisher=Springer-Verlag|location=New York|isbn=978-0-387-94422-7|page=29|edition=1}}</ref> This coincides with the definition given for metric spaces, if <math>(X, d)</math> is a metric space and <math>\tau</math> is the topology generated by <math>d</math>. A limit of a sequence of points <math>\left(x_n\right)_{n \in \N}</math> in a topological space <math>T</math> is a special case of a [[Limit of a function#Functions on topological spaces|limit of a function]]: the [[Domain of a function|domain]] is <math>\N</math> in the space <math>\N \cup \lbrace + \infty \rbrace</math>, with the [[induced topology]] of the [[affinely extended real number system]], the [[Range of a function|range]] is <math>T</math>, and the function argument <math>n</math> tends to <math>+\infty</math>, which in this space is a [[Limit point of a set|limit point]] of <math>\N</math>. ===Properties=== In a [[Hausdorff space]], limits of sequences are unique whenever they exist. This need not be the case in non-Hausdorff spaces; in particular, if two points <math>x</math> and <math>y</math> are [[topologically indistinguishable]], then any sequence that converges to <math>x</math> must converge to <math>y</math> and vice versa.
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