Well-structured transition systems provide the right foundation to compute a
finite basis of the set of predecessors of the upward closure of a state. The
dual problem, to compute a finite representation of the set of successors of
the downward closure of a state, is harder: Until now, the theoretical
framework for manipulating downward-closed sets was missing. We answer this
problem, using insights from domain theory (dcpos and ideal completions), from
topology (sobrifications), and shed new light on the notion of adequate domains
of limits
