The Research On Lower Topological Poset Models Of Topological Spaces

The Scott spaces were defined by D.S.Scott,which referred to the complete lattices endowed with the Scott topology.Later,the Scott topology was also defined in dcpos,and more generally,in general posets.In general,Scott spaces are always T0 but not T1.So,it seems that Scott spaces are too weak in separation to be concerned by classical topologists.Along with a result on Scott spaces appearing,that is every complete metric space is homeomorphic to the maximal point space of a domain equipped with the relative Scott topology,people start focusing on the Scott topology.Moreover,the result reveals that a large class of traditional topological spaces can be represented by the maximal point spaces of some special posets equipped with the relative Scott topology.When investigating the maximal point spaces,the poset consisting of all maximal points is always endowed with the relative Scott topology,however,there is little research on the lower topology.The dissertation is to study the lower topological bounded complete algebraic poset models of T1 topological spaces,the lower topological algebraic domain models of T1 topological spaces,and continuous prequantale models of T1 topological semigroups.The details are listed as follows:In Chapter two,the lower topological poset models of topological spaces are defined.We prove that every T1 topological space has a lower topological bounded complete algebraic poset model.This study shows that a topological space has a lower topological dcpo model if and only if it has a lower topological local dcpo model.Alongside our discussion,the topological properties such as sobriety and well-filteredness of the lower topology on posets are also investigated.In Chapter three,two questions are considered:(1)whether every T1 topological space can have a lower topological domain model;(2)which kind of posets endowed with the lower topology is Choquet complete.And the answer is positive.This study shows that every T1 topological space is homeomorphic to the set of all maximal points of some algebraic domain equipped with the relative lower topology,which may not hold for the Scott topology.We also prove that bounded complete domains endowed with the lower topology are Choquet complete and discuss the lower topological poset models of Tychonoff spaces.The result that every Tychonoff space has a lower topological meet continuous semilattice model is obtained.In Chapter four,the focus is on investigating the continuous prequantale models of topological semigroups.This chapter shows that every T1 topological semigroup satisfying condition(?)can be embedded into a topological semigroup(D,?,?),where D is a domain.Furthermore,by considering the maximal point topological semigroup of a continuous prequantale,it is proven that every T1 topological semigroup satisfying condition(?)has a continuous prequantale model.The continuous poset may not be bounded complete by giving a counterexample.