Well-inside relation ($u\prec v⟺\neg u\vee v=1$)

This relation says that the closure of the open $u$ is contained in $v$. A regular locale has $a={\bigvee }_{u\prec a}u$ for every $a$. If $u\prec a$, then $\neg \neg u\le a$ and $\neg \neg u\prec a$.

A locale is regular when every open is the join of opens well inside it. This expresses regularity without reference to points.