Absorbing System And Its Applications On Function Spaces

Infinite dimensional topology is a vigorous branch in topology. Absorbingsystem is an important tool to research topology structures in infinite dimensionaltopology. Function space is a hot subject of the research in infinitedimensional topology. In this paper, we do some research in Absorbing Systemand its application on making clear structures of function spaces.In Introduction, we give the history of the development of infinite dimensionaltopology and the researching backgroud of this paper.The content of Chapter 1 is Absorbing System Theory. This chapter consistsof 3 sections. In Section 1, we introduce the definition of AbsorbingSystem and its relative concepts. In Section 2, we give some important resultsin Absorbing System Theory. In Section 3, we give one of our results:appling the strongly university property of Absorbing System to characterizethe triple of spaces (Q,∑,c₀), where Q=[-1,1]^w is the Hilbert cube,∑={(x_n)_n∈Q:sup|x_n|＜1} and c₀={(x_n)_n∈∑:(?)=0} aresubspaces of Q.Let (X,ρ) be a metric space. Cld(X) denote all non-empty closed sets inX. For anyε＞0, A(?)X, let B_ρ(A,ε)={y∈X:(?)＜ε}. For anyE,F∈Cld(X), define Hausdorff distanceThen 0≤ρ_H(E,F)≤+∞. Let I= [0,1]. Define an admissable metric d onthe product space X×I as follows: for any (x₁,t₁),(x₂,t₂)∈X×I, putFor any map f:X→I, let Then↓f∈Cld(X×I) if and only if / is upper semi-continuous. Use USC(X)and C(X) to denote the set of all upper semi-continuous maps and continuousmaps from X to I, respectively. For any A C USC(X), putLet USCC(X) = {f∈USC(X):f is compact supported }, CC(X) = C(X)∩USCC(X). Then d_H(↓f,↓g)＜+∞for any f,g∈USCC(X) and hence(↓USCC(X), d_H) is a metric space. Note that if X is compact then USCC(X) =USC(X), CC(X) = C(X).The content of Chapter 2 is the applications of Absorbing System on thefunction spaces on compact spaces. The chapter contains 4 sections. Section 1 ispreliminary. In Section 2, we introduce some results about function spaces. Wegive our research results in Sections 3 and 4. In Section 3, we use our (Q,∑,c₀)Characterization Theorem to prove that for any compact metric space (X,ρ),where |X| denote the cardinal number of X, X' denote the set of all isolatedpoints in X, cl_X denote the closure of a set in X and the symbol "≈" denotesthat there exists a homeomorphism. In Section 4, we use the result in Section 3to prove that (∑,c₀) is an (F_σ,F_σδ)-absorbing system in Q, where F_σ and F_σδdenote the class of all absolute F_σ-sets and the class of all absolute F_σδ-sets,respectively.In Chapter 3, we give our research results on function spaces on noncompactmetric spaces by appling Absorbing Systems. The chapter contains 2sections. In Section 1, we prove that for any metric space (X,ρ), the followingstatements are equivalent:(a).(↓USCC(X),↓CC(X))≈(∑,c₀);(b).↓USCC(X)≈∑; (c).↓USCC(X) is noncompact,σ-compact but could not be an union of countableits finite dimensional closed subspaces;(d). X is noncompact, locally compact, nondiscrete and separable.In Section 2, we consider↓USC(X) and↓C(X) as the subspaces of Cld(?)(X×I).The topology of Cld(?)(X×I) is induced by the metric (?)= min{1, d_H}. Wefirst introduce the results about↓USC(X) that is obtained by Yongjiu Zhang,and then we prove that if (X,ρ) is a locally compact, noncomplete but topologicallycomplete separable metric space with cl_X(X\X')≠X and whosecomplecation X is compact, then↓C(X)≈c₀.