The Katětov construction modified for a -quasi-metric space

We discuss the existence and uniqueness of a T 0 -quasi-metric space q U defined by the following three conditions: (i) q U is bicomplete and supseparable, (ii) every isometry between two finite subspaces of q U extends to an isometry of q U onto itself, and (iii) q U contains an isometric copy of every supseparable T 0 -quasi-metric space.