Approximations of relations by continuous functions

Let X be a Tychonoff space, C ( X ) be the space of all continuous real-valued functions defined on X and CL ( X × R ) be the hyperspace of all nonempty closed subsets of X × R . We prove the following result. Let X be a countably paracompact normal space. The following are equivalent: (a) dim X = 0 ; (b) the closure of C ( X ) in CL ( X × R ) with the Vietoris topology consists of all F ∈ CL ( X × R ) such that F ( x ) ≠ ∅ for every x ∈ X and F maps isolated points into singletons; (c) each usco map which maps isolated points into singletons can be approximated by continuous functions in CL ( X × R ) with the locally finite topology. From the mentioned result we can also obtain the answer to Problem 5.5 in [Lʹ. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173–2182] and to Question 5.5 in [R.A. McCoy, Comparison of hyperspace and function space topologies, Quad. Mat. 3 (1998) 243–258] in the realm of normal, countably paracompact, strongly zero-dimensional spaces. Generalizations of some results from [Lʹ. Holá, R.A. McCoy, Relations approximated by continuous functions, Proc. Amer. Math. Soc. 133 (2005) 2173–2182] are also given.