= Sobrification
{title2=$\eta:X\to\operatorname{Sob}(X),\quad x\mapsto\overline{\{x\}}$}
The sobrification has one point for every nonempty <irreducible closed subset> $A$ of $X$, with opens $\widehat U=\{A:A\cap U\ne\varnothing\}$. Irreducibility gives $\widehat U\cap\widehat V=\widehat{U\cap V}$, and the map from original opens to these opens is bijective, detected by singleton closures. An irreducible closed subset of the new space has complement $\widehat W$; $F=X\setminus W$ is an irreducible closed subset of $X$ and uniquely generates it. Thus the new space is a <sober space>.
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