On di-injective T0-quasi-metric spaces

In this article it is shown that every q-hyperconvex T0-quasi-metric space is di-injective without appealing to Zorn's lemma. We also demonstrate that QX as constructed by Kemajou et al. and Q(X) (the space of all Katětov function pairs on X) are di-injective. Moreover we prove that di-injective T0-quasi-metric spaces do not contain proper essential extensions. Among other results, we state a number of ways in which the di-injective hull can be characterized.