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 (X) of C(X) such that the pair ( (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification (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 × .
@article{bwmeta1.element.doi-10_2478_s11533-009-0041-4,
author = {Atsushi Kogasaka and Katsuro Sakai},
title = {A Hilbert cube compactification of the function space with the compact-open topology},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {670-682},
zbl = {1192.54005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0041-4}
}
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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