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

Hanbiao Yang 1 , Katsuro Sakai 2 , and Katsuhisa Koshino 3
  • 1 School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China
  • 2 Katsuro Sakai: Department of Mathematics, Faculty of Engineering, Kanagawa University, Yokohama, 221-8686, Japan
  • 3 Division of Mathematics, Pure and Applied Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan

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 = I is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

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