Characterising points which make P-frames

A P-frame is a completely regular frame whose cozero part is a Boolean algebra. It is known that these frames are precisely those the points of whose Stone–Čech compactification are all of a certain kind, akin to P-points in Tychonoff spaces. Our principal aim is to characterise such points in all completely regular frames using filters in sublocale lattices. We thus define a filter on (as opposed to “in”) a locale L   to be a filter in the lattice Sℓ(L)Sℓ(L) of sublocales of L, so that its members are sublocales of L and not elements of L. We define convergence for these filters and use that to characterise the points alluded to above. As another application, we also define clustering for these filters and characterise compact locales in terms of both convergence and clustering.