Relations approximated by continuous functions

Holá, L’.; McCoy, R.

doi:10.1090/S0002-9939-05-07793-2

Relations approximated by continuous functions
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by L’. Holá and R. A. McCoy

Proc. Amer. Math. Soc. 133 (2005), 2173-2182

DOI: https://doi.org/10.1090/S0002-9939-05-07793-2

Published electronically: February 15, 2005

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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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