Hanbiao Yang
Metrization of function spaces with the Fell topology

Comment.Math.Univ.Carolin. 53,2 (2012) 307-318.

Abstract: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.

Keywords: space of continuous maps, Fell topology, hyperspace, metrizable, hypograph, separable, first-countable
AMS Subject Classification: 54C35 54E45 54B20