On Noetherian Spaces

A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of well-quasi order, in the sense that an Alexandroff-discrete space is Noetherian iff its specialization quasi-ordering is well. For more general spaces, this opens the way to verifying infinite transition systems based on non-well quasi ordered sets, but where the preimage 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 Nthr of Noetherian spaces is finitely complete and finitely cocomplete. Finally, we note that if X is a Noetherian space, then the set of all (even infinite) subsets of X is again Noetherian, a result that fails for well-quasi orders.