Hyperconvexity in partial metric spaces: challenges and outlooks

TL;DR

This paper explores various definitions of hyperconvexity in partial metric spaces, highlighting the limitations of classical notions and the unique challenges in extending these concepts.

Contribution

It introduces multiple approaches to hyperconvexity in partial metric spaces and analyzes their properties, revealing fundamental differences from classical metric spaces.

Findings

Classical hyperconvexity notions do not fully transfer to partial metric spaces

The Aronszajn--Panitchpakdi hyperconvexity analogue fails key properties

Different definitions of hyperconvexity exhibit distinct behaviors in partial metric spaces

Abstract

In this article, we present several different ways to define hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn--Panitchpakdi notion of hyperconvexity fails to exhibit certain key properties present in the classical metric setting.