= Convergence of a sequence in a topological space
{title2=$x_n\to x$}
A <sequence> $(x_n)$ in a <topological space> converges to $x$ if every <open> <neighbourhood> of $x$ contains all sufficiently late terms:
$$
\forall U\text{ open with }x\in U\quad\exists N\quad\forall n\geq N:\ x_n\in U.
$$
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.<Point-Set Topology by Romyar Sharifi/Chapter 1: Topological spaces>[Definition 1.5.5, Proposition 1.5.11 and Proposition 1.6.20: neighbourhood convergence, uniqueness and the metric formulation]{ref}
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