【作者】 李慧；

【导师】 李庆国；

【作者基本信息】 湖南大学 ， 数学， 2020， 博士

【摘要】 Scott空间是指完备格带有Scott拓扑的拓扑空间(后来Scott拓扑也定义在dcpo上,更一般地,定义在一般偏序集上).一般情况下,Scott空间是T0的而不是T1的,起初由于Scott空间在分离性上太弱了而不能引起拓扑学家的关注.然而随着A.Edalat和R.Heckmann证明的每一个完备度量空间都可以同胚于一个带着Scott拓扑的domain的极大点空间这一结果的出现,人们开始把焦点放在Scott空间上.同时,这个结果说明有一类拓扑空间可以用带着Scott拓扑的特殊偏序集的极大点空间来表示.之前一般在研究极大点空间时,极大点集带有的拓扑都是相对Scott拓扑,而对于极大点集带有下拓扑的研究却很少.本文主要研究了 T1拓扑空间的带有下拓扑的有界完备的代数偏序集模型和带有下拓扑的代数domain模型以及T1拓扑半群的连续的prequantale模型.具体内容如下:在第2章中,我们提出了带有下拓扑的偏序集模型的概念,并证明了每一个T1拓扑空间都有一个带有下拓扑的有界完备的代数偏序集模型.我们还证明了一个拓扑空间有一个带有下拓扑的dcpo模型当且仅当它有一个带有下拓扑的局部dcpo模型.另外,我们还讨论了带有下拓扑的偏序集的拓扑性质,比如:sober性和良滤性.在第3章中,我们考虑了两个问题:(1)是否每一个T1拓扑空间都可以有一个带有下拓扑的domain模型;(2)什么样的偏序集带着下拓扑是Choquet完备的.在这一章中我们给出了答案.我们证明了每一个T1拓扑空间都有一个带有下拓扑的代数domain模型,这一结论对于Scott拓扑是不成立的.我们还证明了带有下拓扑的有界完备domain是Choquet完备的,并讨论了 Tychonoff空间的带有下拓扑的偏序集模型,证明了每一个Tychonoff空间都有一个带有下拓扑的交连续半格模型.在第4章中,我们主要探讨拓扑半群的连续prequantale模型.在这一章中我们证明了每一个满足条件(△)的T1拓扑半群都可以嵌入到一个拓扑半群(D,σ,☉)中,其中D是一个domain.然后通过考虑连续的prequantale的极大点拓扑半群,我们证明了每一个满足条件(△)的T1拓扑半群都有一个连续的prequantale模型并举例说明这个连续偏序集不一定是有界完备的.更多还原

【Abstract】 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(Δ)can be embedded into a topological semigroup(D,σ,☉),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(Δ)has a continuous prequantale model.The continuous poset may not be bounded complete by giving a counterexample.更多还原