  We show that in complete metric spaces, $4$-hyperconvexity is equivalent to
finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric
space is $n$-hyperconvex. This generalizes among others results of
Lindenstrauss and answers questions of Aronszajn-Panitchpakdi.
  Furthermore, we prove local-to-global results for externally and weakly
externally hyperconvex subsets of hyperconvex metric spaces and find sufficient
conditions in order for those classes of subsets to be convex with respect to a
geodesic bicombing.
