The Countably S₂-Quasicontinuous Posets

Basing on the ideals used to generalize quasicontinuous posets to general-ized countably approximating posets and S₂-quasicontinuous posets,we introduce the concept of countably S₂-quasicontinuous posets as a common generalization of countably approximating posets and S₂-quasicontinuous posets by using the normal completion operator.We investigate the properties of countably S₂-quasicontinuous posets systematically.The main results are listed as following.We give the definition of countably way-below relation on a general poset and introduce the concept of a countably S₂-quasicontinuous poset.We prove that the countably way-below relation on a countably S₂-quasicontinuous poset satisfies the interpolation property.We obtain that a poset is countably S₂-quasicontinuous iff its weak countable scott topology is locally hypercompact and that the weak countable scott topology on a countably S₂-quasicontinuous poset is countably sober iff the poset is a countably approxi-mating poset.We introduce the concept of a countably S₂-meet continuous poset and show that a poset is countably S₂-continuous if and only if it is both countably S₂-quasicontinuous andS₂-meet continuous.Finally,we give the definition of GS*-convergence class and obtain the characterization of countably S₂-quasicontinuous posets by nets,i.e.,GS*-convergence is topological in a poset iff the poset is count-able S₂-quasicontinuous.