More on Externally <i>q</i>-Hyperconvex Subsets of <i>T</i><sub>0</sub>-Quasi-Metric Spaces

We continue earlier research on T0-quasi-metric spaces which are externally q-hyperconvex. We focus on external q-hyperconvex subsets of T0-quasi-metric spaces in particular. We demonstrate that a countable family of pairwise intersecting externally q-hyperconvex subsets has a non-empty intersection that is external q-hyperconvex under specific requirements on the underlying space (see Proposition 22). Last but not least, we demonstrate that if A is a subset of a supseparable and externally q-hyperconvex space Y, where Y ⊆ X, then A is also externally q-hyperconvex in X (Proposition 25).