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

Key Points

• The aim is to explore the properties of countably sober spaces and establish duality in countably directed complete posets.
• Defined equivalent conditions for countably sober, ?*-well-filtered, and ?*-d-spaces.
• Proved the relationship between first countable P-spaces and ?*-d-spaces.
• Established a topological duality for countably directed complete posets using C-sobriety.
• A first countable P-space is countably sober if and only if it is an ?*-d-space.
• Equivalences among defined properties give insights into the structure of these posets.