Hausdorff Spaces

A Hausdorff orspace is a topological space in which distinct points have disjoint neighbourhoods, implying that any two points are part of disjoint open sets.are Hausdorff with the usual topology because if thenso each ofcan be placed inside open balls of radius

The Hausdorff condition implies the uniqueness of limits of functions, since ifis continuous anda sequence convergent to x so thatandare subsequences ofthen forgiven there existssuch thatimplies

Take

Sinceis convergent and f is continuous there exists such thatand similarly impliesHence

Hence

This is a contradiction hence limits of functions are unique. Sinceis also a sequence, limits of sequences are unique.

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