Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let [...] be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair [...] , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.
@article{bwmeta1.element.doi-10_1515_math-2015-0021,
author = {Hanbiao Yang and Katsuro Sakai and Katsuhisa Koshino},
title = {A function space from a compact metrizable space to a dendrite with the hypo-graph topology},
journal = {Open Mathematics},
volume = {13},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0021}
}
Hanbiao Yang; Katsuro Sakai; Katsuhisa Koshino. A function space from a compact metrizable space to a dendrite with the hypo-graph topology. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0021/
[1] Banakh T., Radul T., Zarichnyi M., Absorbing Sets in Infinite-Dimensional Manifolds, Math. Stud. Monogr. Ser., 1, VNTL Publishers, Lviv, 1996. | Zbl 1147.54322
[2] Bing R.H., Partitioning a set, Bull. Amer. Math. Soc., 1949, 55, 1101-1110. | Zbl 0036.11702
[3] Borsuk K., Theory of Retracts, MM, 44, Polish Sci. Publ., Warsaw, 1966.
[4] Koshino K., Infinite-dimensional manifolds and their pairs, Ph.D. thesis, University of Tsukuba, 2014.
[5] Koshino K., Sakai K., A Hilbert cube compactification of a function space from a Peano space into a one-dimensional locally compact absolute retract, Topology Appl. 161 (2014), 37-57.[WoS] | Zbl 1286.54010
[6] van Mill J., Infinite-Dimensional Topology, Prerequisites and Introduction, North-Holland Math. Library, 43, Elsevier Sci. Publ., Amsterdam, 1989.
[7] Moise E.E., Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math. Soc., 1949, 55, 1111-1121. | Zbl 0036.11801
[8] Moise E.E., A note of correction, Proc. Amer. Math. Soc., 1951, 2, 838.
[9] Sakai K., The completions of metric ANR’s and homotopy dense subsets, J. Math. Soc. Japan, 2000, 52, 835-846. | Zbl 0974.57013
[10] Sakai K., Geometric Aspects of General Topology, SMM, Springer, Tokyo, 2013. | Zbl 1280.54001
[11] Sakai K., Uehara S., A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 1999, 92, 107-118.[WoS] | Zbl 0926.54008
[12] Torunczyk H., On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math., 1980, 106, 31-40. | Zbl 0346.57004
[13] Yang Z., The hyperspace of the regions below of continuous maps is homeomorphic to c0, Topology Appl., 2006, 153(15), 2908-2921. | Zbl 1111.54008
[14] Yang Z., Zhou X., A pair of spaces of upper semi-continuous maps and continuous maps, Topology Appl., 2007, 154(8), 1737-1747.[WoS] | Zbl 1119.54010
[15] Whyburn G.T., Analytic Topology, AMS Colloq. Publ., 28, Amer. Math. Soc., Providence, R.I., 1963.