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Relations approximated by continuous functions

Authors: L'. Holá and R. A. McCoy
Journal: Proc. Amer. Math. Soc. 133 (2005), 2173-2182
MSC (2000): Primary 54C35, 54B20, 54C08
Published electronically: February 15, 2005
MathSciNet review: 2137885
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Abstract: Let $X$ be a Tychonoff space, let $C(X)$ be the space of all continuous real-valued functions defined on $X$ and let $CL(X \times R)$ be the hyperspace of all nonempty closed subsets of $X\times R$. We prove the following result. Let $X$ be a locally connected, countably paracompact, normal $q$-space without isolated points, and let $F \in CL(X \times R)$. Then $F$ is in the closure of $C(X)$ in $CL(X \times R)$ with the locally finite topology if and only if $F$is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

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Additional Information

L'. Holá
Affiliation: Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

R. A. McCoy
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Keywords: Set-valued mapping, Vietoris topology, locally finite topology, upper-semicontinuous multifunction, usco map, cusco map
Received by editor(s): October 14, 2003
Received by editor(s) in revised form: April 8, 2004
Published electronically: February 15, 2005
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society

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