= Connected subsets of a cofinite space
For a <set> $X$ with its <cofinite topology>, a <subset> $S$ is a <connected subset> exactly when $S$ is empty, a <singleton>, or infinite. Indeed, its <subspace topology> is the <cofinite topology> on $S$: the traces of <open sets> are precisely $S\setminus F$ for finite $F\subseteq S$, together with the <empty set>. For infinite $S$, two nonempty <open sets> intersect, because their <complements> have finite <union>. Hence $S$ is a <hyperconnected space> and therefore a <connected space>. A finite $S$ has the <discrete topology>, which admits a <separation of a topological space> as soon as $S$ has at least two points.
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