Three different ways of describing Priestley spaces are presented: as the objects of a category which arises in the equivalence induced by an adjunction F . U : OrdTopop ¡æ Lat, as limits of (suitable) finite topologically-discrete preordered spaces (i.e. as profinite preorders) and as the 2-compact ordered spaces, in the sense of Engelking and Mr¢¥owka [5], three situations where, for discrete-ordered topological spaces, one obtains Stone spaces instead of Priestley spaces.

FCT/Centro de Matemática da Universidade de Coimbra