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## Open Mathematics

2015 | 13 | 1 |
Tytuł artykułu

### A function space from a compact metrizable space to a dendrite with the hypo-graph topology

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Języki publikacji
EN
Abstrakty
EN
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}.
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EN
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Tom
Numer
Opis fizyczny
Daty
otrzymano
2013-09-17
zaakceptowano
2014-12-15
online
2015-03-04
Twórcy
autor
• School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China, hongsejulebu@sina.com
autor
• Katsuro Sakai: Department of Mathematics, Faculty of Engineering, Kanagawa University, Yokohama, 221-8686, Japan, k.sakai.top@gmail.com
autor
Bibliografia
• [1] Banakh T., Radul T., Zarichnyi M., Absorbing Sets in Infinite-Dimensional Manifolds, Math. Stud. Monogr. Ser., 1, VNTL Publishers, Lviv, 1996.
• [2] Bing R.H., Partitioning a set, Bull. Amer. Math. Soc., 1949, 55, 1101-1110.
• [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]
• [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.
• [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.
• [10] Sakai K., Geometric Aspects of General Topology, SMM, Springer, Tokyo, 2013.
• [11] Sakai K., Uehara S., A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 1999, 92, 107-118.[WoS]
• [12] Torunczyk H., On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math., 1980, 106, 31-40.
• [13] Yang Z., The hyperspace of the regions below of continuous maps is homeomorphic to c0, Topology Appl., 2006, 153(15), 2908-2921.
• [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]
• [15] Whyburn G.T., Analytic Topology, AMS Colloq. Publ., 28, Amer. Math. Soc., Providence, R.I., 1963.
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