The Topological Structure Of Function Spaces And Hyperspaces Of The Regions Below Of Functions

In this content, we mainly proved that some function spaces and hyper-spaces of the regions below of functions are homeomorphic to some widely used infinitely model spaces.Let (X, d) be a metric space. A single real-valued function f: Xâ†'R is upper semi-continuous if f-1(-∞, t) is open in X for each t∈R. The function f is called a Lipschitz map, if there are some k≥0 such that|f(x)-f(x')|≤k·d(x, x') for all x,x'∈X. The Lipschitz constant of f is defined by lipf inf{k≥0:|f(x)-f(x')|≤k·d(x,x')}. Suppose that X is also a polyhedron, the above f is called a piece-wise linear map, if there is a subdivision X'of X, such that f|c is linear for each cell C∈X'.We use USC(X), C(X), LIP(X) and PL(X) to represent the families of all the upper semi-continuous, continuous, Lipschitz and piecewise functions from X to I=[0,1], respectively. Besides, we also consider the sets k-LIP(X) {f∈LIP(X):lipf≤k}, and LIPk(X)={f∈LIP(X):lipf<k}.The mainly discussed infinitely dimensional model spaces in this paper are:Q=[0, 1]ω, s=(0, 1)ω, E={(xi)∈s:supi|xi|<1} andσ={(xi)∈s: xi=0 except for finitly many i}.In Chapter three, we mainly discussed the topological structure of function spaces.Let X be a non-compact, locally compact, separable metric space, endowed the sets C(X), LIP(X), k-LIP(X) and LIPk(X) with compact-open topology, we proved that, the pair (C(X), LIP(X)) is homeomorphic to (s,∑). In the same conditions, we also proved that (k-LIP(X), LIPk(X))≈(Q,∑) for each k>0.In Chapter four, we mainly discussed the topological structure of hyper-spaces of the regions below of functions.For the metric space (X, d), denote by Cld(X×I) the set of all non-empty closed set in the product space X×I. We can endowed the set Cld(X×I) with Hausdorff metric topology and Fell topology to make it to be a topology space, denote by CldH(X x I) and CldF(X x I) respectively.For each f∈USC(X), let Note that f∈USC(X) if and only if↓f is a closed subset of the product space X x I. We use↓USC(X) to denote the set{↓f:f E USC(X)}, then↓USC(X) can inherit the topologies of the hyperspaces CldH(X x I) and CldF(X x I), and denoted by↓USCH(X) and↓USCF(X) respectively. The sets↓C(X),↓LIP(X) and↓PL(X) can be similarly defined. After↓USC(X) is endowed with some topology, these sets can inherit the topology from↓USC(X) as its subspace.First, We proved that if X is a non-zero dimensional compact Euclidean polyhedron, then and for each k>0,Besides, We also showed that if X is a non-compact, locally compact, separable metrizable space, we have and for each k>0,...