Convergence of a sequence in a topological space
ID: convergence-of-a-sequence-in-a-topological-space
A sequence in a topological space converges to if every open neighbourhood of contains all sufficiently late terms:For a metric space, this is the usual limit of a sequence condition expressed through open balls. For a Hausdorff space, distinct proposed limits have disjoint open neighbourhoods, so a sequence cannot eventually stay in both and the limit is unique. The sequence convergence in the cofinite topology demonstrates what can change when that separation property is absent.[ref]
Chapter 1: Topological spaces. Definition 1.5.5, Proposition 1.5.11 and Proposition 1.6.20: neighbourhood convergence, uniqueness and the metric formulation
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