A Hilbert cube compactification of the Banach space of continuous functions

Abstract

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 × R ̄ ) of nonempty compact subsets of X × R ̄ endowed with the Vietoris topology, where R ̄ = [−∞, ∞] is the extended real line (cf. (Fedorchuk, 1991)). Then the closure R ̄ (X) of C( X) in exp(X × R ̄ ) is a compactification of C( X). We show that the pair ( C ̄ (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).