Core compactness of ordered topological spaces

In this paper, we investigate the property of core compactness of ordered topological spaces. Particularly, we give a series of characterizations of the core compactness for directed spaces. Several results obtained in this paper are closely related to a long-standing open problem in Open problems in Topology (J. van Mill, G. M. Reed Eds., North-Holland, 1990): Which distributive continuous lattice's spectrum is exactly a sober locally compact Scott space?

Automated summary

This paper investigates the property of core compactness in ordered topological spaces, with a particular focus on directed spaces. A series of characterizations of core compactness in directed spaces are presented. The results obtained in this paper are closely related to a long-standing open problem in the field of Topology.