summary: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.
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机译：摘要：对于Tychonoff空间$ X $，令$ \ downarrow {\ rm C} _F（X）$为所有具有Fell拓扑的从$ X $到$ [0,1] $的连续图的象形图族。证明如果$ \ downarrow {\ rm C} _F（X）$是可量化的，则$ X $具有一个密集的可分离的可量化的局部紧凑开放子集。此外，对于第一个可数空间$ X $，当且仅当$ X $本身是局部紧凑的可分离空间时，才可以对$ \ downarrow {\ rm C} _F（X）$进行度量。存在Tychonoff空间$ X $，使得$ \ downarrow {\ rm C} _F（X）$是可度量的，但是$ X $不可首数。
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