Trying to come up with a point-free formulation of the $T_1$ axiom I thought of the following condition on a frame/locale $L$:
$$\forall U \in L \, . \, U \wedge \bigwedge \{ V \in L : U \vee V = \top \} = \bot$$
It turns out that this condition is weaker than $T_1$ for spatial locales since it holds eg for $T_1$ spaces with non-$T_1$ sobrification like $\mathbb N$ with the cofinite topology, but at least it seems to exclude non-trivial Alexandroff locales.
Does anybody know what this condition is called?