ω*-d-spaces, C-sobriety and duality of countably directed complete posets

C-sobriety characterizes the P-spaces which are determined by their open sets lattices. In this paper, on the one hand, we obtain the equivalent definitions of countably sober, ?*-well-filtered and ?*-d-spaces, and prove that a first countable P-space X is countably sober if and only if it is an ?*-d-space. On the other hand, we establish a topological duality for countably directed complete posets via C-sobriety.