Research On H-directivity And SI-compact Space

Directed sets are a very important notion in domain theory.They play an important role in domain theory.This thesis introduces a generalized directed sets,called H-directed sets.Using H-directed sets,we define the notions of H-continous posets and H-topological spaces,and study their properties.Compactness plays an important role in topology,and attract the attentions of many scholars.In 2015,D.Zhao and W.K.Ho introduced a new kind of compactness,called SI-compactness.They proved that every SI-compact topological space is always compact.They asked whether every compact space is SI-compact.Based on this problem,we studied some properties of D(P),where D(P)denotes the poset of all nonempty compact saturated subsets of the Scott space of P equipped with the reverse inclusion order.By using Isbell's non-sober complete lattice,we also prove that there exists a compact topological space which is not SI-compact.The main content of this thesis is arranged as follows:Chapter One:Preliminaries.This chapter recalls the basic notions and related knowledges of domain theory and lattice theory.Chapter Two:H-directed sets and H-topological spaces.Firstly,we introduces the notions of H-directed sets and H-continuous posets,and prove that H-way below relation satisfies the interpolation property.We also present a sufficient condition of continuous posets being H-continuous posets.Secondly,we introduced the notions of H-topological spaces,and study some properties of H-topological spaces on H-continuous dcpos.We prove that every H-topological space on H-continuous dcpo is sober.Finally,we introduced the concept of H-convergence,and prove that if L is a strong hcpo,then H-convergence is topological if and only if L is an Hcontinuous poset.Chapter Three:Compact spaces and SI-compact spaces.Firstly,we introduced the concept of H-compactness,and proved that compactness is equivalent to H-compactness.We also prove that if X is a compact space and ?(O(X))is a sober space,then X is SI-compact.Secondly,We discuss some properties of(D(L),?),and prove that for a complete lattices L,(D(L),?)is order isomorphic to the open set lattice of some topological space.We also prove that if P,Q are domains(Tlattices or their Scott spaces being Co-sober spaces),then P is order isomorphic to Q if and only if(D,(P),?)is order isomorphic to(D(Q),?)).Finally,using Isbell's non-sober complete lattice,we show that there is a compact space which is not SIcompact.By using the sobrification of topological spaces,we obtain that there is a sober compact space which fails to be SI-compact.