For a space "X", ψ: "X" → pt(Ω("X")) is a homeomorphism if and only if it is bijective. Using the characterization via meet-prime elements of the open set lattice, one sees that this is the case if and only if every meet-prime open set is of the form "X" \ Cl{"x"} for a unique "x". Alternatively, every join-prime closed set is the closure of a unique point, where "join-prime" can be replaced by (join-) irreducible since we are in a distributive lattice. Spaces with this property are called sober.