Topological classification of function spaces with the Fell topology IV.

For a Tychonoff space X , let ↓ C F ( 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 ↓ C F ( X ) is homeomorphic to c 0 ∪ ( Q ∖ Σ ) if ↓ C F ( 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 = { ( x n ) ∈ Q : sup ⁡ | x n | < 1 } and c 0 = { ( x n ) ∈ Σ : lim ⁡ x n = 0 } are its subspaces. Combining results in the previous papers of the series, we give the topological classification for all metrizable function spaces ↓ C F ( X ) of k-spaces X . [ABSTRACT FROM AUTHOR]