Convergence

Conditions for the uniqueness of the limit

Let be a topological space. If is Hausdorff then every convergent sequence has a unique limit, topology i.e. every sequence has at most one limit point. Moreover, if is first-countable, then it is hausdorff iff all limits are unique.

Proof

First assume is Hausdorff. Let in , and some . Then there exist open neighbourhood and such that . Moreover there exists , such that for all . Then for all such and hence . Thus limits are unique for any hausdorff space, without invoking first-countability.

Now assume is first-countable with unique limit, and let such that . Let and be nested open neighbourhood bases of and respectively. Assume is not hausdorff, i.e. for all . Then we can construct a sequence in such that for all , in which case and violating the uniqueness of limits. Therefore must be hausdorff.

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