Topological classification of function spaces with the Fell topology II

For a Tychonoff space X  , let ↓CF(X)↓CF(X) denote the collection of the hypographs of all continuous maps from X   to [0,1][0,1] with the Fell topology. We show that for a Tychonoff k-space X  , ↓CF(X)↓CF(X) is homeomorphic to c0c0 if and only if ↓CF(X)↓CF(X) is metrizable and not Baire if and only if X   is a weakly locally compact and hemicompact ℵ0ℵ0-space without dense set of isolated points, where Q=[−1,1]ωQ=[−1,1]ω is the Hilbert cube, Σ={(xn)∈Q:sup⁡|xn|<1}Σ={(xn)∈Q:sup⁡|xn|<1} and c0={(xn)∈Σ:lim⁡xn=0}c0={(xn)∈Σ:lim⁡xn=0} are the subspaces of it.