Topological structure of the space of lower semi-continuous functions

Let L(X) be the space of all lower semi-continuous extended real-valued functions on a Hausdorff space X, where, by identifying each f with the epi-graph epi(f), L(X) is regarded the subspace of the space Cld(*)(X x R) of all closed sets in X x R with the Fell topology. Let LSC(X) = {f is an element of L(X) vertical bar f(X) boolean AND R not equal theta, f(X) subset of (-infinity, infinity]} and LSCB(X) = {f is an element of L(X) vertical bar f(X) is a bounded subset of R}. We show that L(X) is homeomorphic to the Hilbert cube Q = [-1, 1](N) if and only if X is second countable, locally compact and infinite. In this case, it is proved that (L(X), LSC(X), LSCB (X)) is homeomorphic to (Cone Q, Q x (0, 1), Sigma x (0, 1)) (resp. (Q, s, Sigma)) if X is compact (resp. X is non-compact), where Cone Q = (Q x I)/(Q x {1}) is the cone over Q, s = (-1, 1)(N) is the pseudo-interior, Sigma = {(xi)i is an element of N E Q vertical bar suPi is an element of N vertical bar xi vertical bar < 1} is the radial-interior.