Topological classification of function spaces with the Fell topology III

For a Tychonoff space X  , denote ↓CF(X)↓CF(X) to be the set of hypographs of all continuous functions from X   to [0,1][0,1] equipped with the Fell topology. We proveMain Theorem. Let X be a Tychonoff space satisfying the following conditions:(i)↓CF(X)↓CF(X)is metrizable;(ii)X is a non-discrete k-space;(iii)X has a dense set of isolated points;(iv)the set of non-isolated points of X is not compact.Then  ↓CF(X)↓CF(X)is homeomorphic to the subspace  c0∪(Q∖Σ)c0∪(Q∖Σ)of Q, where  Q=[−1,1]NQ=[−1,1]Nis the Hilbert cube,  Σ={(xn)∈Q:supn∈N⁡|xn|<1}and  c0={(xn)∈Σ:limn→∞⁡xn=0}.