 Sobrification

The sobrification has one point for every nonempty irreducible closed subset $A$ of $X$, with opens $\hat{U}=\{A:A\cap U\ne \varnothing \}$. Irreducibility gives $\hat{U}\cap \hat{V}=\widehat{U\cap V}$, 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 $\hat{W}$; $F=X\setminus W$ is an irreducible closed subset of $X$ and uniquely generates it. Thus the new space is a sober space.