Scott spaces and sober spaces (Q1962094)

It is well known that every \(T_0\) topology on a set induces a partial ordering (the specialization ordering) on the same set. Furthermore, if the topology is sober, then the partial ordering has directed joins, and the topology is contained in the Scott topology for the induced ordering (though the Scott topology need not be sober in general). In this paper the author investigates what happens to the notion of sobriety if you jettison the \(T_0\) axiom, and what happens to the Scott topology if you jettison the assumption that the underlying pre-order (is a partial order and) has directed joins. As is to be expected, there is some fragmentation of conditions which are equivalent under the assumptions thus jettisoned, but there turns out to be a fair amount that can still be said.