The notion of hyperconvexity of metric spaces was introduced by Aronszajn and Panitchpakdi in 1956 in their study of extensions of uniformly continuous mappings between metric spaces. From the topological point of view, a hyperconvex space is an absolute retract via a nonexpansive retraction. By the theorem of Nachbin and Kelley, hyperconvex real Banach spaces can be treated as Stonian spaces C(K) of all real-valued continuous functions on extremally disconnected compact Hausdorff spaces K.

On the other hand, the notion of a partial metric space was introduced by Matthews in 1994. He showed, roughly speaking, how metric-like tools can be extended to non-Hausdorff topologies, and he also indicated some applications of this class of spaces in the study of the denotational semantics of programming languages. Further applications of partial metrics can also be found in the geometry of Banach spaces.

In this talk, we present several different approaches to defining hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn–Panitchpakdi notion of hyperconvexity fails to possess certain key properties present in the classical metric setting. Finally, we outline some perspectives for further research.