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 一类非紧度量空间上的连续函数空间 Spaces of Continuous Maps on a Class of Noncompact Metric Spaces 吴拿达1,杨忠强2 WU Nada1,*, YANG Zhongqiang2 1. 韩山师范学院数学与应用数学系，潮州，广东, 5210412. 汕头大学数学系，汕头，广东, 515063 1. Department of Mathematics and Applied Mathematics, Hanshan Normal University, Chaozhou, Guangdong, 521041, P. R. China;2. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, P. R. China 收稿日期: 2011-05-03 出版日期: 2013-08-25 DOI: 10.11845/sxjz.20130413b
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 摘要 对一个度量空间$(X,\rho)$, 设 $\dd{\rm C}(X)$ 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$X\times I$ 上的 Hausdorff 度量诱导出的拓扑. 本文证明了下面的结果：如果 $(X, \rho)$ 是一个非紧的、 局部紧的、可分的、 完全有界的度量空间, 则 $\dd{\rm C}(X)$ 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$c_0=\{(x_n)_{n\in\N}\in [-1,1]^{\omega}: {\rm sup}|x_n|<1$且$\lim_{n\to +\infty}x_n=0\}$. 特别地, 对赋予通常度量的开区间$(0,1)$, $\dd{\rm C}((0,1))$ 同胚于$c_0$. 关键词 ： 连续函数,  非紧空间,  Hausdorff 拓扑,  吸收子 Abstract：For a metric space $(X,\rho)$, let $\dd{\rm C}(X)$ 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 $X\times I$. In the present paper, the following result is proved: if $(X, \rho)$ is a noncompact, locally compact, totally bounded, separable metric space, then $\dd{\rm C}(X)$ is homeomorphic to $c_0$ if and only if the set of all isolated points of $X$ is not dense in $X$, where $c_0=\{(x_n)_{n\in\N}\in [-1,1]^{\omega}: {\rm sup}|x_n|<1$ and $\lim_{n\to +\infty}x_n=0\}$. Specially, for the open interval $(0,1)$ with the usual metric, $\dd{\rm C}((0,1))$ is homeomorphic to $c_0$. Key words： noncompact space    Hausdorff topology    absorber 基金资助:Supported by NSFC (No. 10971125), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20094402110001) and Hanshan Normal University Start-up Project for Ph. D. (No. QD20091202). 通讯作者: * ndwu@hstc.edu.cn
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