Discussion For Several Problems Related To Well-filterifications Of T₀ Spaces

In this dissertation,some problems related to well-filterifications of T0 spaces are considered,including the following aspects:First,we introduce an equivalence relation on the collection of KF-sets.Then we define a topology on the collection of equivalent classes.We obtain a concrete characterization of the well-filterifications of T0 spaces by transfinite induction.Second,we investigate the KF-set of Ershov’s basic construction Z.We introduce the concept of wf-rank,which is an ordinal that measures how far a T0 space is from being a well-filtered space.Using the co-finite topology on N and transfinite induction,we prove that for any ordinal α,there is a T0 space whose wf-rank equals α.Additionally,we give an answer to the problem proposed by Xiaoquan Xu,and prove that the collection of Rudin sets is not equal to the collection of WD-sets.Third,we focus on the sufficient conditions and necessary conditions for certain properties of topological spaces which are preserved in passing for function space.The following are the main results:(ⅰ)if[X,Y]is a d-space for some nonempty space X,then Y is a d-space;(ⅱ)for any non-empty space X,if[X,Y]is well-filtered,then Y is well-filtered;(ⅲ)for any core compact space X,if Y is well-filtered,then[X,Y]is well-filtered;(ⅳ)if[X,Y]is an injective space,resp.a densely injective space for some non-empty space X,then Y is also an injective space,resp.a densely injective space;(ⅴ)for any non-empty Δ-space X,if[X,Y]is a Δ-space,then Y is a Δ-space.Finally,we investigate the Hofmann-Mislove Theorem for ω-well-filtered spaces using the countably generated Scott-open filters,and prove that the collection of countably generated Scott-open filters is isomorphic to the compact saturated subsets.Some conditions for dual Hofmann-Mislove Theorem of ω-well-filtered spaces are also investigated by generated topology.