一类非紧度量空间上的连续函数空间（英文）

Spaces of Continuous Maps on a Class of Noncompact Metric Spaces

对一个度量空间(X,ρ),设↓C(X)是从X到I=[0,1]的连续函数下方图形全体之集赋予由度量空间X×I上的Hausdorff度量诱导出的拓扑.本文证明了下面的结果:如果(X,ρ)是一个非紧的、局部紧的、可分的、完全有界的度量空间,则↓C(X)同胚于c_0当且仅当X上的孤立点全体之集在X中不稠密,这里c_0={(xn)n∈N∈[-1,1]ω:sup|x+n|<1且lim_(n→+∞)x_n=0}.特别地,对赋予通常度量的开区间(0,1),↓C((0,1))同胚于c_0.

Abstract： For a metric space （X, p）, let ↓C（X） be the family of hypographs of all continuous maps from X to I= [0,1] endowed with the topology induced by the Hausdorff metric of the metric space X × I. In the present paper, the following result is proved： if （X, p） is a noncompact, locally compact, totally bounded, separable metric space, then ↓C（X） is home- omorphic to co if and only if the set of all isolated points of X is not dense in X, where co =c0={（xn）n∈N∈[-1,1]^ω： sup[xn] 〈 1 and limn-＋∞ xn = 0}. Specially, for the open interval （0, 1） with the usual metric, ↓C（（0, 1）） is homeomorphic to co.