Atomic topology
= Atomic topology
{title2=$J(U)=\{S\subseteq yU:S\ne\varnothing\}$}
The atomic topology declares exactly the nonempty sieves covering. It exists precisely when arrows with a common codomain admit a common refinement. On nonempty finite sets and surjections this follows from the fiber product. Its J-closed sieves are only the empty and maximal sieves.