A Study On Closed Interval Posets And Some Function Spaces

Topology mainly studies the topological invariance of space under homeomor-phic mapping,in which separability are important topic and used to describe the re-lationship between disjoint sets or different points.In years,T₀spaces with poor sep-arability had seldom natural applications.People did not think T₀spaces with poor separability are much significance to research until the emergence of domain theory,which also links topology with computer science.Then,T₀spaces with poor sepa-rability gradually attracts people’s attention.The Scott topology,proposed by Scott and defined on posets,is a kind of natural T₀space.In contrast,metric spaces are with very good separation properties.Edalat and Heckmann established some con-nections between metric spaces and domain theory by maximal point spaces of well-behaved posets.For a metric space（X,d）,they define a continuous poset（BX,（?））concerned with formal bolls such that X is homeomorphic to the set Max（BX）of all maximal points of BX,where Max（BX）is endowed with the relative Scott topology.Given that the relative Scott topologies on maximum point Spaces of domains are homeomorphic to some topological spaces with better separability.Therefore,domain theory is naturally inextricably linked with metric space.Topological prop-erties of function space are important in many branches of mathematics.In par-ticular,several statements of Ascoli’s theorem give several necessary and sufficient conditions for determining which subsets are compact in some function spaces with some typical topologies.With the development of infinite dimensional topology,fruitful topological structures of these function spaces have been discovered.Be-cause of too many function spaces,people recognized it is necessary to classify func-tion spaces according to topological homeomorphism.Among function spaces with good separability,some are homeomorphic to the famous infinite dimensional topo-logical space-Hilbert space Q,some are homeomorphic to some subsets,such asΣ,c₀of Q,and some are homeomorphic to the separable Hilbert space （?）₂.For more than half a century,many scholars at home and abroad have obtained abundant results in the study of function space and hyperspace.In this paper we first study a series of properties of closed interval posets with the inverse inclusion order and obtain a necessary and sufficient condition that poset has closed interval domain.In the following,the concept of quasiexactness is pro-posed and some properties related to weak domain and quasi continuous domain are studied.Besides,the topological structure of function space of transitive map-pings in dynamical systems is studied.Chapter 1 mainly introduces the development of domain theory and infinite dimensional function space,and some connections be-tween the two disciplines,as well as their significance to other disciplines.We consider closed interval posets IP with the inverse inclusion order for posets P.Firstly,we discuss some properties of the way-below relation on IP.Sec-ondly,a necessary and sufficient conditions for IP to be a domain is given.Using this result,it is proved that there is a domain model consisting of closed intervals for any poset satisfying that condition.These results generalize ones proved by Martin and Panangaden.Then,we define the bi-Scott continuous mapping,and give two sufficient conditions for the Scott continuous extension of the bi-Scott continuous mapping between two maximal point spaces.In addition,several topologies related to closed intervals and the relationships among these topologies are discussed.We also consider some weaker properties related to weak domains and quasi continuous domains are considered,and common generalization of these two do-mains is also proposed,which is called a quasiexact dcpo.First,we introduce the weak way-below relation between two non-empty subsets of a poset,then define quasiexact poset.Secondly,some relations between quasiexact poset,quasi-continuous domain and weak domain are discussed.In addition,the weakly way-below finite-deterministic topology is introduced,and some connections among the way-below topology and the Scott topology,as well as the weakly way-below topology first pro-posed by Mushburn,are studied.We also show that for any dcpo,if it is quasiexact,moderately continuous and weak way-below relation weakly increasing,then it is domain.This result is very similar to characterizing domains with meet continuity and quasi-continuity.Finally,weak way-below relation on closed interval posets are considered,and a condition on posets is given for the corresponding closed interval posets to be domains.For function spaces,we first prove a sufficient condition for metric spaces to be ARs.Let C（I）be the set of all continuous self-maps on I=[0,1]with the topology of uniformly convergence.A map f∈C（I）is called transitive if for every pair of non-empty open sets U,V in I,there exists a positive integer n such that U∩f^-n（V）=（?）.The set of all transitive maps and its closure in the space C（I）are denoted by T（I）and T（I）,respectively.By some results of Kolyada et al.,we show a main theorem:Both subspaces T（I）and T（I）are homeomorphic to the separable Hilbert space（?）₂.This result is more precise than those results obtained by Kolyada et al.[34].As the corollaries,we obtain two results.There exists a homeomorphism h:C（I）→I×（?）₂such that h（T（I））={0}×（?）₂.Moreover,For every open cover U of T（I）,there exists a homeomorphism h:T（I）→T（I）such that(id_T（I）,h)（?）U,that is,for every f∈T（I）,there exists U∈U such that f,h（f）∈U.