Lusin and Suslin properties of function spaces

A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For aTychonoff space X letCp( X), Ck( X) andC.F( X) be the space of continuous realvalued functions on X, endowed with the topology of pointwise convergence, the compactopen topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is s-compact, (2) Cp( X) is Suslin, (3) Ck ( X) is Suslin, (4) C.F( X) is Suslin, (5) Cp(X) is Lusin, (6) Ck ( X) is Lusin, (7) C.F( X) is Lusin, (8) Cp(X) is Fs -Lusin, (9) Ck (X) is Fs -Lusin, (10) C.F(X) is Cds -Lusin. Also we construct an example of a sequential.0-space X with a unique non-isolated point such that the function spaces Cp( X), Ck ( X) and C. F( X) are non-Suslin.