Inspired by classic work of Wallman and more recent work of Jung-Kegelmann-Moshier and Vickers , we show how to encode general subbases of stably locally compact spaces via certain entailment relations. We further build this up to a categorical duality encompassing the classic Priestley-Stone duality and its various extensions to stably locally compact spaces by Shirota, De Vries, Hofmann-Lawson (in the stable case), Jung-Sünderhauf, Hansoul-Poussart, Bezhanishvili-Jansana, van Gool and Bice-Starling.

We characterize Yoneda completeness for non-symmetric distances by combinations of metric and directed completeness. One of these generalizes the Kostanek–Waszkiewicz theorem on formal balls.