CSI-Continuous Spaces And Countably K-Bounded Sober Spaces

In non-Hausdorff spaces and domain theory,irreducible sets and directed sets are two important subsets.The definition of continuous domains relies on directed sets,and S I-continuous spaces and sober spaces are defined via irreducible sets.In this thesis,replacing irreducible sets by countably irreducible sets in the definition of S I-continuous spaces(resp.,k-bounded sober spaces),we introduce the concept of CSI-continuous spaces(resp.,countably k-bounded sober spaces).We try to discuss some properties of CSI-continuous spaces and countably k-bounded spaces systematically.We introduce the topology,which we call the CSI topology,induced by their countably irreducible sets,and discuss its basic properties.We prove that if the topological space is a P-space,then its CSI-topological space is a C-space if and only if the topological space is a CSI-continuous space.In addition,some characterizations of CSI-continuous spaces are given,and we prove that if the upper power space of a topological space is a countably k-bounded sober space,then the topological space is a countably k-bounded sober space,and countably k-bounded sober spaces are heritable with respect to saturated subspaces.We also show that the image of countably k-bounded sober spaces are still countably k-bounded sober spaces under the special continuous mappings.