A discussion of well-filteredness and sobriety

Abstract

It remains an open problem as to whether a core-compact dcpo is sober. We give a sufficient condition for a core-compact dcpo to be sober. We introduce and study a new kind of posets—join continuous posets. Meanwhile, we investigate the relationships among sober, well-filtered and weak well-filtered topological properties. Based on the above results, we give some characterizations for a poset to be Lawson compact. Moreover, we have also solved the open problems posed by Xu and Zhao. Finally, we have considerably simplified the counterexample given by Isbell based on the work of Goubault in his blog.