Quasicontinuity of posets via Scott topology and sobrification
The authors introduce and study quasicontinuous posets that are not directed complete and generalize various known results from domain theory to this context. For example, standard results about quasicontinuous dcpos are generalized to quasicontinuous posets, e.g., a poset is quasicontinuous if and only if its lattice of Scott open sets is hypercontinuous and a quasicontinuous poset is meet-continuous if and only if it is a continuous poset. By way of generalization from the theory of continuous posets, it is also shown that the directed completion of a quasicontinuous poset is again quasicontinuous and equipped with the Scott topology is the sobrification of the Scott topology of the original poset. A slightly more general version of the last result was obtained by the reviewer [Topology Appl. 21, 73--76 (1985; Zbl 0575.54013)].
scientific article; zbMATH DE number 5146180
scientific article