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July 2015 The Baire property of certain hypo-graph spaces
Katsuhisa Koshino
Tsukuba J. Math. 39(1): 29-38 (July 2015). DOI: 10.21099/tkbjm/1438951816


Let $X$ be a compact metrizable space and $Y$ be a nondegenerate dendrite with an end point 0. For each continuous function $f : X \rightarrow Y$, we define the hypo-graph $\downarrow f = \cup_{x \in X \{x} \times [0, f(x]$ of $f$, where $[0, f(x)]$ is the unique path from 0 to $f(x)$ in $Y$. Then we can regard $\downarrow \mathrm{C}(X,Y) = \{\downarrow f | f : X \rightarrow Y$ is continuous} as a subspace of the hyperspace consisting of non-empty closed sets in $X \times Y$ equipped with the Vietoris topology. In this paper, we prove that $\downarrow \mathrm{C}(X,Y)$ is a Baire space if and only if the set of isolated points of $X$ is dense.


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Katsuhisa Koshino. "The Baire property of certain hypo-graph spaces." Tsukuba J. Math. 39 (1) 29 - 38, July 2015.


Published: July 2015
First available in Project Euclid: 7 August 2015

zbMATH: 1330.54026
MathSciNet: MR3383877
Digital Object Identifier: 10.21099/tkbjm/1438951816

Primary: 54C35‎
Secondary: 54B20 , 54E52

Keywords: Baire space , dendrite , function space , hyperspace , hypo-graph , the Hausdor. metric , the Vietoris topology

Rights: Copyright © 2015 University of Tsukuba, Institute of Mathematics


Vol.39 • No. 1 • July 2015
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