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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 4, 2015

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

Hanbiao Yang, Katsuro Sakai and Katsuhisa Koshino
From the journal Open Mathematics

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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Received: 2013-9-17
Accepted: 2014-12-15
Published Online: 2015-3-4

© 2015 Yang Hanbiao et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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