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

Hanbiao Yang; Katsuro Sakai; Katsuhisa Koshino

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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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}.

How to cite

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Hanbiao Yang, Katsuro Sakai, and Katsuhisa Koshino. "A function space from a compact metrizable space to a dendrite with the hypo-graph topology." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/269971>.

@article{HanbiaoYang2015,
abstract = {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\}.},
author = {Hanbiao Yang, Katsuro Sakai, Katsuhisa Koshino},
journal = {Open Mathematics},
keywords = {Function space; Hyperspace; Hypo-graph; The Vietoris topology; Dendrite; The Hilbert cube},
language = {eng},
number = {1},
pages = {null},
title = {A function space from a compact metrizable space to a dendrite with the hypo-graph topology},
url = {http://eudml.org/doc/269971},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Hanbiao Yang
AU - Katsuro Sakai
AU - Katsuhisa Koshino
TI - A function space from a compact metrizable space to a dendrite with the hypo-graph topology
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - 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}.
LA - eng
KW - Function space; Hyperspace; Hypo-graph; The Vietoris topology; Dendrite; The Hilbert cube
UR - http://eudml.org/doc/269971
ER -

References

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  10. [10] Sakai K., Geometric Aspects of General Topology, SMM, Springer, Tokyo, 2013. Zbl1280.54001
  11. [11] Sakai K., Uehara S., A Hilbert cube compactification of the Banach space of continuous functions, Topology Appl., 1999, 92, 107-118.[WoS] Zbl0926.54008
  12. [12] Torunczyk H., On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. Math., 1980, 106, 31-40. Zbl0346.57004
  13. [13] Yang Z., The hyperspace of the regions below of continuous maps is homeomorphic to c0, Topology Appl., 2006, 153(15), 2908-2921. Zbl1111.54008
  14. [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] Zbl1119.54010
  15. [15] Whyburn G.T., Analytic Topology, AMS Colloq. Publ., 28, Amer. Math. Soc., Providence, R.I., 1963. 

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