Alexandroo and Scott Topologies for Generalized Ultrametric Spaces

Both preorders and ordinary ultrametric spaces are instances of generalized ul-trametric spaces. Every generalized ultrametric space can be isometrically embedded in a (complete) function space by means of an ultrametric version of the categorical Yoneda Lemma. This simple fact gives naturally rise to: 1. a topology for generalized ultrametric spaces which for arbitrary preorders corresponds to the Alexandroo topology and for ordinary ultrametric spaces reduces to the-ball topology; 2. a topology for algebraic complete generalized ultrametric spaces generalizing both the Scott topology for arbitrary algebraic complete partial orders and the-ball topology for complete ultrametric spaces.