Join-continuity + Hypercontinuity = Prime continuity

Abstract

A remarkable result due to Kou, Liu & Luo states that the condition of continuity for a dcpo can be split into quasi-continuity and meet-continuity. Their argument contained a gap, however, which is probably why the authors of the monograph Continuous Lattices and Domains used a different (and fairly sophisticated) sequence of lemmas in order to establish the result. In this note we show that by considering the Stone dual, that is, the lattice of Scott-open subsets, a straightforward proof may be given. We do this by showing that a complete lattice is prime-continuous if and only if it is join-continuous and hypercontinuous. A pleasant side effect of this approach is that the characterisation of continuity by Kou, Liu & Luo also holds for posets, not just dcpos.