The Property Of The Scott Topology Of Directed Complete Posets

Domain theory was developed by Turing Prize winner Dana Scott,to provide a suitable denotational semantic model for computer functional programming languages by using order structures.The intersection of order and topology is one of the features of domain theory.Among them,the sober separability of the directed complete posets endowed with the Scott topology(we also call it the Scott space of directed complete posets)is a hot topic in domain theory,which has attracted the attention of many scholars.This paper revolves around the sober separability of the directed complete posets,studying the property of the Scott topology of directed complete posets that is an implicit topology,and solves several open problems about sober separability.The details are as follows:In Chapter 2,in order to solve the problem:"Are core-compact dcpos sober?",we introduce the concept of join-continuous posets,and give an approximate answer to the above problem based on this concept.Meanwhile,we investigate the relationships among sober,well-filtered and weak well-filtered topological properties.Based on the above results,we give some characterizations for a poset to be Lawson compact.Moreover,we also solve the open problems posed by Xu and Zhao.Finally,we have considerably simplified the counterexample given by Isbell based on the work of Goubault-Larrecq in his blog.In Chapter 3,we answer two problems concerning sobriety.The first problem:whether there is a topological space,such that its closed set lattice has a non-sober Scott topology.The second problem:whether the category of allk-bounded sober spaces reflective in the category of all T0 spaces and continuous mappings that preserve all existing sups of irreducible sets.We also offer some sufficient conditions for a dcpo equipped with the Scott topology to be k-bounded well-filtered.In Chapter 4,we show that the core compactly generated spaces are closed underω-well-filterification and D-completion.Furthermore,we find that the core compactly generated topology of the Smyth power space of a well-filtered space coincides with the Scott topology induced by its specialization order.Finally,we provide a characterization for core-compactness of core compactly generated spaces.In Chapter 5,first,we offer an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set.This also gives a negative answer to the problem proposed by Xu,since each tapered closed set is a closed WD-set.Next,we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements.Finally,for a given T0 space,we give some sufficient conditions which guarantee that each pair of its standard D-completion,standard wellfilterification and standard sobrification agrees.In Chapter 6,we answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober.This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space.In addition,we prove that the topology of the product space ΣP×ΣQ coincides with the Scott topology of the product poset P × Q if the set Id(P)and Id(Q)of all non-trivial ideals of posets P and Q are both countable.Based on this,it is deduced that a directed complete poset P has a sober Scott space,if Id(P)is countable and ΣP is coherent and well-filtered.In particular,every complete lattice L with Id(L)countable has a sober Scott space.