非紧度量空间上的上半连续函数下方图形超空间

Hyperspaces of the Regions Below of Upper Semi-continuous Maps on Non-compact Metric Spaces

设X=(X,d)是一个度量空间,↓USC(X)表示从X到单位闭区间I=[0,1]上所有上半连续函数下方图形的集合并且赋予Hausdoff度量拓扑.本文证明了:如果X是一个连通的、完备的、非紧的度量空间,则↓USC(X)同胚于权为2^(w(X))的Hilbert空间,这里w(X)表示X的权.如果X是一个拓扑完备的、非紧的度量空间并且X的完备化是紧的,则↓USC(X)同胚于Hilbert空间l_2.

For a metric space（X,d）,↓USC（X） denotes the family of the regions below of all upper semi-continuous maps from X to I =[0,1].We consider the space↓USC（X） topologized by the Hausdoff metric.In this paper,we show that if X is a connected,complete, non-compact metric space,then↓USC（X） is homeomorphic to the non-separable Hilbert space whose weight is 2^（ω（x））,whereω（X） denotes the weight of X;if X is a topologically complete, non-compact metric space and the completion of X is compact,then↓USC（X） is homeomorphic to the Hilbert space l_2.