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A Hilbert cube compactification of the Banach space of continuous functions

✍ Scribed by Katsuro Sakai; Shigenori Uehara

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
787 KB
Volume
92
Category
Article
ISSN
0166-8641

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✦ Synopsis

Let C(X) be the Banach space of continuous real-valued functions of an infinite compacturn X with the sup-norm, which is homeomorphic to the pseudo-interior s = (-I, I)"' of the Hilbert cube Q = [-1, llw. We can regard C(X) as a subspace of the hyperspace exp(X x E) of nonempty compact subsets of X x E endowed with the Vietoris topology, where E = [-cx), ~1 is the extended real line (cf. (Fedorchuk, 1991)). Then the closure c(X) of C(X) in exp(X x E) 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).

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