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Ranks for baire multifunctions

Pandelis Dodos (2003)

Colloquium Mathematicae

Various ordinal ranks for Baire-1 real-valued functions, which have been used in the literature, are adapted to provide ranks for Baire-1 multifunctions. A new rank is also introduced which, roughly speaking, gives an estimate of how far a Baire-1 multifunction is from being upper semicontinuous.

Relations approximated by continuous functions in the Vietoris topology

L'. Holá, R. A. McCoy (2007)

Fundamenta Mathematicae

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...

Relaxation theorem for set-valued functions with decomposable values

Andrzej Kisielewicz (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Let (T,F,μ) be a separable probability measure space with a nonatomic measure μ. A subset K ⊂ L(T,Rⁿ) is said to be decomposable if for every A ∈ F and f ∈ K, g ∈ K one has f χ A + g χ T K . Using the property of decomposability as a substitute for convexity a relaxation theorem for fixed point sets of set-valued function is given.

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