Reductions On Equivalence Relations Generated By Universal Sets

Borel reduction is a fundamental concept in Descriptive set theory,we often use it to compare the complexity of different equivalence relations.But usually,for equiv-alence relations E and F,there can be no Borel reductions between them.So,if we still want to reduce the two equivalence relations,we have to consider some more com-plicated reductions.Let X,Y be Polish spaces,?(?)(Y),A(?)C X×Y.A is called a universal for ? if {Ax:x ? X} = ?,and Ax = {y:(x,y)?A)? A}.For any set A(?)X × Y,we define an equivalence relation EA as xEAx'(?)Ax Ax'.We will mainly show the following results in this thesis:(1)If A is a?n1 set universal for all nonempty closed subsets of Y,then EA is a?(?n1)equivalence relation and Ea ??(?n1)id(2?).(2)If A is a ?11 set universal for all countable subsets of Y,then EA is ?(?11)equivalence relation,and(?)EA??(?11)=+ and =+??21EA;(?)if V = L,then EA ??21 id(2?);(?)for n ? 2,if every ?ni set is Lebesgue measurable or has the Baire property,then EA(?)?n1 id(2?).(iv)for n ? 2,if every ?n1 set has the Baire property,and E is any ?30 equivalence relation,then EA(?)?n1 E.We arrange the structure of this dissertation as follows:The first chapter is the introduction.In this chapter,we will give a brief intro-duction on the background of this dissertation,then we will give some fundamental knowledge in classical descriptive set theory and invariant descriptive set theory,such as basic definitions about equivalence relations and reducibilities.Finally,we will list the main results we obtained in this dissertation.The second chapter is the results about the reducibility between EA and some other typical equivalence relations,where A is a universal for the closed sets of an uncountable Polish space X.The third chapter is the results about the reducibility and nonreducibility between EA and some other typical equivalence relations,where A is a universal for the countable sets of an uncountable Polish space X.The forth chapter is the summary.In this chapter,we summarize the main points of this dissertation and put forward some other closely related and unsolved problems.