Geodesic
Metrization of function spaces with the Fell topology
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 2, pp. 307-318.

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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.
Classification : 54B20, 54C35, 54E45
Keywords: space of continuous maps; Fell topology; hyperspace; metrizable; hypograph; separable; first-countable 
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Yang, Hanbiao. Metrization of function spaces with the Fell topology. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 2, pp. 307-318. http://geodesic.mathdoc.fr/item/CMUC_2012__53_2_a9/