Well Topologies

Call a topology well if and only if every open is compact. The starting point of this note is that this notion generalizes that of well-quasi order, in the sense that an Alexandroff topology is well if and only if its specialisation quasi-ordering is well. For more general topologies, this opens the way to verifying infinite transition systems based on non-well quasi ordered sets, but where the Pre operator satisfies an additional continuity assumption. The technical development rests heavily on techniques arising from topology and domain theory, including sobriety and the de Groot dual of a stably compact space. We show that the category Well of well topological spaces is finitely complete and finitely cocomplete. Finally, we note that if X is well topologized, then the set of all (even infinite) subsets of X is again well topologized, a result that fails for well-quasi orders.