Displaying similar documents to “A Hilbert cube compactification of the function space with the compact-open topology”

Relations approximated by continuous functions in the Vietoris topology

L'. Holá, R. A. McCoy (2007)

Fundamenta Mathematicae

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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...