Countably Sober Spaces

From the point of separation in topological spaces,sobriety is between To and T2,and being T1 and being sober are incomparable properties.Sober spaces have wonderful properties and play an important role in domain theory.In this thesis,we introduce the concept of countably sober spaces as a kind of generalization of sober spaces,and discuss some properties of countably sober spaces systematically.We introduce the concept of countably irreducible sets,discuss some basic prop-erties of them,and give the definition of countably sober spaces.We discuss some topological constructions preserve countably sobriety.In particularly,we prove that the category CSOB with countably sober spaces and continuous mappings is a com-plete category.By using completely prime countable filters,Scott open countable filters,etc.,we give some characterizations of countably sober spaces.We prove that the a-Scott on a countably approximating poset is countably sober,and we obtain the Hofmann-Mislove Theorem for countably sober spaces.