A Hilbert cube compactification of the function space with the compact-open topology
Atsushi Kogasaka ; Katsuro Sakai
Open Mathematics, Tome 7 (2009), p. 670-682 / Harvested from The Polish Digital Mathematics Library

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 Fréchet 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¯ (X) of C(X) such that the pair (C¯ (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification C¯ (X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

Publié le : 2009-01-01
EUDML-ID : urn:eudml:doc:269213
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     title = {A Hilbert cube compactification of the function space with the compact-open topology},
     journal = {Open Mathematics},
     volume = {7},
     year = {2009},
     pages = {670-682},
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     language = {en},
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Atsushi Kogasaka; Katsuro Sakai. A Hilbert cube compactification of the function space with the compact-open topology. Open Mathematics, Tome 7 (2009) pp. 670-682. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0041-4/

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