Metrization of function spaces with the Fell topology

Commentationes mathematicae Universitatis Carolinae

Hanbiao Yang¹ 

关键词: space of continuous maps;    Fell topology;    hyperspace;    metrizable;    hypograph;    separable;    first-countable;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

For a Tychonoff space $X$, let $\downarrow \!{\rm C}_F(X)$ be the family of hypographs of all continuous maps from $X$ to $[0,1]$ endowed with the Fell topology. It is proved that $X$ has a dense separable metrizable locally compact open subset if $\downarrow \!{\rm C}_F(X)$ is metrizable. Moreover, for a first-countable space $X$, $\downarrow \!{\rm C}_F(X)$ is metrizable if and only if $X$ itself is a locally compact separable metrizable space. There exists a Tychonoff space $X$ such that $\downarrow \!{\rm C}_F(X)$ is metrizable but $X$ is not first-countable.

【 授权许可】

CC BY