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Relations approximated by continuous functions in the Vietoris topology

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Holá L., McCoy R.

Fundamenta Mathematicae

2007

tom 195

nr 3

205-219

Let X be a Tikhonov 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 locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ X, F(x) is a singleton set; (4) F is upper semicontinuous; and (5) F forces local semiboundedness. This gives an answer to Problem 5.5 in [HM] and to Question 5.5 in [Mc2] in the realm of locally connected locally compact paracompact spaces.

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1 Fundamenta Mathematicae

1 Holá L.

1 McCoy R.

1 2007

Wyniki wyszukiwania

Relations approximated by continuous functions in the Vietoris topology