Countably Sober Spaces

As a generalization of sober spaces, we introduce the concept of countably sober spaces and prove that some topological constructions preserve countable sobriety. In particularly, we prove that the category with countably sober spaces and continuous mappings is a complete category. We give some characterizations of countably sober spaces via countable filters and obtain the Hofmann-Mislove Theorem for countably sober spaces.