Does anyone know of any work that might be relevant to the following question: Given a (non-sober) topology T on a set X, is T always expressible as the intersection of the sober topologies which contain it? Equivalently, given a set S \subseteq X which is not open in T, can we find a sober topology T' \supseteq T in which S is still not open? It's true when T is the cofinite topology (topologies that contain the cofinite topology are just the T_1 topologies, of course), and (slightly less obviously) when T is the indiscrete topology. It also seems to work for most of the familiar examples of non-sober topologies, but the methods I've used to prove it in each case are pretty ad hoc. A positive answer to the question (at least in the case when T is a T_0 topology) would be useful in characterizing the soluble sublocales of a spatial locale, about which I spoke in Cape Town last month. Peter Johnstone