Abstract
For a Tychonoff space X, we use ↓USC F (X) and ↓C F (X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0, 1] with the subspace topologies of the hyperspace Cld F (X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology. In this paper, we shall show that there exists a homeomorphism h: ↓USC F (X) → Q = [−1, 1]ω such that h(↓C F (X)) = c 0 = {(x n ) ∈ Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.
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References
Curtis, D. W., Schori, R. M.: Hyperspaces of Peano continua are Hilbert cube. Fund. Math., 101, 19–38 (1978)
Wojdyslawski, M.: Rétractes absoulus et hyperespaces des continus. Fund. Math., 32, 184–192 (1939)
Van Mill, J.: The Infinite-dimensional Topology of Function Spaces, North-Holland Math. Library, Vol. 64, Elsevier Sci. Publ. B.V., Amsterdam, 2001
Sakai, K., Yang, Z.: Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube. Topology Appl., 127, 331–342 (2003)
Yang, Z.: The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification. Sci. China Ser. A, 48, 469–484 (2005)
Yang, Z.: The hyperspace of the regions below of continuous maps is homeomorphic to c 0. Topology Appl., 153, 2908–2921 (2006)
Yang, Z., Zhou, X.: A pair of spaces of upper semi-continuous maps and continuous maps. Topology Appl., 154, 1737–1747 (2007)
Yang, Z., Wu, N.: Topological position of the set of continuous maps in the set of upper semi-continuous maps. Sci. China Ser. A, 52, 1815–1828 (2009)
Sakai, K., Uehara, S.: Topological structure of the space of lower semi-continuous functions. Comment. Math. Univ. Carolina, 47(1), 113–126 (2006)
Beer, G.: Topologies on Closed and Closed Convex Sets, Kluwer Acad. Publ., Dordrecht, 1993
Van Mill, J.: The Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library, Vol. 43, Elsevier Sci. Publ. B.V., Amsterdam, 1989
Baars, J., Gladdines, H., Van Mill, J.: Absorbing systems in infinite-dimensional manifold. Topology Appl., 50, 147–182 (1993)
Dijkstra, J. J., Van Mill, J., Mogilski, J.: The space of infinite-dimensional compacta and other topological copy’s of (l2 f)ω. Pacific. J. Math., 152, 255–273 (1992)
Ogura, Y.: On some metrics compatible with the Fell-Matheron topology. Internat. J. Approx. Reason., 46, 65–73 (2007)
Sakai, K.: The completions of metric ANR’s and homotopy dense subsets. J.Math. Soc. Japan, 52, 835–846 (2000)
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Supported by National Natural Science Foundation of China (Grant No. 10971125)
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Yang, Z.Q., Zhang, B.C. The hyperspace of the regions below continuous maps with the fell topology. Acta. Math. Sin.-English Ser. 28, 57–66 (2012). https://doi.org/10.1007/s10114-012-0030-6
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DOI: https://doi.org/10.1007/s10114-012-0030-6