Topological classification of function spaces with the Fell topology IV

For a Tychonoff space X, let ↓CF(X) denote the collection of the hypographs of all continuous maps from X to [0,1] with the Fell topology. We show that, for a Tychonoff non-discrete k-space X, the function space ↓CF(X) is homeomorphic to c0∪(Q∖Σ) if ↓CF(X) is metrizable and the set of isolated points of X is dense in X, where Q=[−1,1]N is the Hilbert cube, Σ={(xn)∈Q:sup⁡|xn|<1} and c0={(xn)∈Σ:lim⁡xn=0} are its subspaces. Combining results in the previous papers of the series, we give the topological classification for all metrizable function spaces ↓CF(X) of k-spaces X.