Categories of Locally Hypercompact Spaces and Quasicontinuous Posets

A subset of a topological space is hypercompact if its saturation (the intersection of its neighborhoods) is generated by a finite set. Locally hypercompact spaces are defined by the existence of hypercompact neighborhood bases at each point. We exhibit many useful properties of such spaces, often based on Rudin’s Lemma, which is equivalent to the Ultrafilter Principle and ensures that the Scott spaces of quasicontinuous domains are exactly the locally hypercompact sober spaces. We characterize their patch spaces (the Lawson spaces) as hyperconvex and hyperregular pospaces in which every monotone net has a supremum to which it converges. Moreover, we find extensions to the non-sober case by replacing suprema with cuts, and we provide topological generalizations of known facts for quasicontinuous posets. Similar results are obtained for hypercompactly based spaces and quasialgebraic posets. Furthermore, locally hypercompact spaces are described by certain relations between finite sets and points, providing a quasiuniform approach to such spaces. Our results lead to diverse old and new equivalences and dualities for categories of locally hypercompact spaces or quasicontinuous posets.