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 USCC F (X,
) of all upper semi-continuous set-valued functions φ: X →
= [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCC F (X,
) is inherited from the Fell hyperspace Cld* F (X ×
) of all closed sets in X ×
.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beer G., Topologies on closed and closed convex sets, Kluwer Acad. Publ., Dordrecht, 1993
Beer G., On the Fell topology, Set-Valued Anal., 1993, 1, 69–80
Bessaga C., Pełczyński A., Selected topics in infinite-dimensional topology, MM 58, Polish Sci. Publ., Warsaw, 1975
Bing R.H., Partioning a set, Bull. Amer. Math. Soc., 1949, 55, 1101–1110
Chapman T.A., Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc., 1971, 154, 399–426
Engelking R., General topology (Revised and complete edition), Sigma Ser. in Pure Math., Heldermann Verlag, Berlin, 1989
Fedorchuk V.V., On certain topological properties of completions of function spaces with respect to Hausdorff uniformity, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, 77–80 (in Russian); English translation: Moscow Univ. Math. Bull., 1991, 46, 56–58
Fedorchuk V.V., Completions of spaces of functions on compact spaces with respect to the Hausdorff uniformity, Tr. Semin. im. I. G. Petrovskogo, 1995, 18, 213–235 (in Russian); English translation: J. of Math. Sci., 1996, 80, 2118–2129
Fell J.M.G., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc., 1962, 13, 472–476
Moise E.E., Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math. Soc., 1949, 55, 1111–1121
van Mill J., Infinite-dimensional topology, Elsevier Sci. Publ. B.V., Amsterdam, 1989
Sakai K., Uehara S., A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 1999, 92, 107–118
Sakai K., Yang Z., Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube, Topology Appl., 2002, 127, 331–342
Tominaga A., One some properties of non-compact Peano spaces, J. Sci. Hiroshima Univ., Ser.A, 1956, 19(3), 457–467
Tominaga A., Tanaka T., Convexification of locally connected generalized continua, J. Sci. Hiroshima Univ., Ser.A, 1955, 19(2), 301–306
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kogasaka, A., Sakai, K. A Hilbert cube compactification of the function space with the compact-open topology. centr.eur.j.math. 7, 670–682 (2009). https://doi.org/10.2478/s11533-009-0041-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-009-0041-4