A Hilbert cube compactification of the function space with the compact-open topology

Atsushi Kogasaka; Katsuro Sakai

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 670-682
  • ISSN: 2391-5455

Abstract

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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 × .

How to cite

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Atsushi Kogasaka, and Katsuro Sakai. "A Hilbert cube compactification of the function space with the compact-open topology." Open Mathematics 7.4 (2009): 670-682. <http://eudml.org/doc/269213>.

@article{AtsushiKogasaka2009,
abstract = {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 \[ \bar\{C\} \] (X) of C(X) such that the pair (\[ \bar\{C\} \] (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification \[ \bar\{C\} \] (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 × .},
author = {Atsushi Kogasaka, Katsuro Sakai},
journal = {Open Mathematics},
keywords = {The Hilbert cube; The psuedo-interior; The psuedo-boundary; Compactification; Function space; The compact-open topology; The Fell topology; Hilbert cube; pseudo-interior; pseudo-boundary; compactification; function space; compact-open topology; Fell topology},
language = {eng},
number = {4},
pages = {670-682},
title = {A Hilbert cube compactification of the function space with the compact-open topology},
url = {http://eudml.org/doc/269213},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Atsushi Kogasaka
AU - Katsuro Sakai
TI - A Hilbert cube compactification of the function space with the compact-open topology
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 670
EP - 682
AB - 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 \[ \bar{C} \] (X) of C(X) such that the pair (\[ \bar{C} \] (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification \[ \bar{C} \] (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 × .
LA - eng
KW - The Hilbert cube; The psuedo-interior; The psuedo-boundary; Compactification; Function space; The compact-open topology; The Fell topology; Hilbert cube; pseudo-interior; pseudo-boundary; compactification; function space; compact-open topology; Fell topology
UR - http://eudml.org/doc/269213
ER -

References

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