Complementary open and closed local operators ($o(U)\wedge c(U)=\mathrm{id},o(U)\vee c(U)=\top$)

The open local operator and closed local operator associated with $U$ have meet the identity and join the largest local operator. The meet identity is $(u\Rightarrow p)\wedge (u\vee p)=p$. Every mono factors through its union with $U$ as a closed-operator-dense mono followed by an open-operator-dense mono, proving that their join makes every mono dense.