A Hilbert cube compactification of the Banach space of continuous functions

Let C(X) be the Banach space of continuous real-valued functions of an infinite compactum X with the sup-norm, which is homeomorphic to the pseudo-interior s = (−1, 1)ω of the Hilbert cube Q = [−1, 1]ω. We can regard C(X) as a subspace of the hyperspace exp(X × R̄) of nonempty compact subsets of X × R̄ endowed with the Vietoris topology, where R̄ = [−∞, ∞] is the extended real line (cf. (Fedorchuk, 1991)). Then the closure R̄(X) of C(X) in exp(X × R̄) is a compactification of C(X). We show that the pair (C̄(X), C(X)) is homeomorphic to (Q, s) if X is locally connected. As a corollary, we give the affirmative answer to a question of Fedorchuk (Fedorchuk, 1996, Question 2.6).