A Hilbert cube compactification of the Banach space of continuous functions
Let \(C(X)\) be the Banach space of continuous real-valued functions of a compact metric space \(X\) with the sup-norm. Identifying a map \(f:X\to \mathbb{R} \subset \overline{\mathbb{R}}\) with its graph, one can regard \(C(X)\) as a subspace of \(\exp (X\times \overline{\mathbb{R}})\) (the hyperspace of nonempty compact subsets of \(X\times \overline{\mathbb{R}}\) endowed with the Vietoris topology) [\textit{V. V. Fedorchuk}, Mosc. Univ. Math. Bull. 46, No. 4, 56-58 (1991); translation from Vestn. Mosk. Univ., Ser. I 1991, No. 4, 77-80 (1991; Zbl 0783.54016)]. Then the authors consider the closure \(\overline{C}(X)\) of \(C(X)\) in \(\exp (X\times \overline{\mathbb{R}})\), that is a compactification of \(C(X)\). The main result of the paper is: if \(X\) is a locally connected infinite compact metric space, then \((\overline{C}(X), C(X))\) is homeomorphic to \((Q,s)\), where \(Q\) is the Hilbert cube \([-1,1]^\omega\) and \(s= (-1,1)^\omega\). As a corollary, the authors give an affirmative answer to question 2.6 of a paper by \textit{V. V. Fedorchuk} [J. Math. Sci., New York 80, No. 5, 2118-2129 (1996); translation from Tr. Semin. Im. I. G. Petrovskogo 18, 213-235 (1995; Zbl 0857.54020)]. Finally, in an Appendix, they give a counterexample to a gap in the proof of the proposition 2.3 of that paper.
scientific article; zbMATH DE number 1307876