A Hilbert cube compactification of the function space with the compact-open topology

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Frechet space, so it is homeomorphic to the psuedo-interior s = (-1, 1)(N) of the Hilbert cube Q = [-1, 1](N). In this paper, generalizing the Sakai-Uehara's result to the non-compact case, we construct a natural compactification (C) over bar (X) of C(X) such that the pair ((C) over bar (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification (C) over bar (X) coincides with the space USCCF(X, (R) over bar) of all upper semi-continuous set-valued functions phi : X -> (R) over bar = [-infinity, infinity] such that each phi(x) is a closed interval, where the topology for USCCF(X, (R) over bar) is inherited from the Fell hyperspace Cld(F)*(X x (R) over bar) of all closed in X x (R) over bar.