 A topology may be defined on \(D\) by selecting certain increasing sets as the opens. More precisely, if \(I\) is an ideal and \(C\) is a cone, then \(C\) is open iff \(C\cap I \ne\emptyset\) provided that \(\bigvee I\in C\), that is, provided that the supremum of \(I\) is an element of \(C\).  (For example, complements of principal ideals are open.) \(f\) turns out to be continuous in the usual topological sense, that is, the inverse image of an open set is open, when \(D\) and \(E\) are taken together with their topologies. This motivates the earlier labeling of these functions as continuous. Notably, all continuous functions are monotone.