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James boundaries and σ-fragmented selectors

B. Cascales, M. Muñoz, J. Orihuela (2008)

Studia Mathematica

We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for B X * we study different kinds of conditions on T, besides T being countable, which ensure that X * = s p a n T ¯ | | · | | . (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if J : X → 2 B X * is the duality mapping and there exists a σ-fragmented map f: X → X* such that B(f(x),ϵ)...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation f = ∑ C ∈ a C φ C with a C ∈ E and a partition of unity φ C : C ∈ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction f | P attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...

Mesocompactness and selection theory

Peng-fei Yan, Zhongqiang Yang (2012)

Commentationes Mathematicae Universitatis Carolinae

A topological space X is called mesocompact (sequentially mesocompact) if for every open cover 𝒰 of X , there exists an open refinement 𝒱 of 𝒰 such that { V ∈ 𝒱 : V ∩ K ≠ ∅ } is finite for every compact set (converging sequence including its limit point) K in X . In this paper, we give some characterizations of mesocompact (sequentially mesocompact) spaces using selection theory.

Michael's theorem for Lipschitz cells in o-minimal structures

Małgorzata Czapla, Wiesław Pawłucki (2016)

Annales Polonici Mathematici

A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.

Multifunctions of two variables: examples and counterexamples

Jürgen Appell (1996)

Banach Center Publications

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

On a question of E.A. Michael

Vladimir V. Filippov (2004)

Commentationes Mathematicae Universitatis Carolinae

A negative answer to a question of E.A. Michael is given: A convex G δ -subset Y of a Hilbert space is constructed together with a l.s.c. map Y → Y having closed convex values and no continuous selection.

On a selection theorem of Blum and Swaminathan

Takamitsu Yamauchi (2004)

Commentationes Mathematicae Universitatis Carolinae

Blum and Swaminathan [Pacific J. Math. 93 (1981), 251–260] introduced the notion of ℬ -fixedness for set-valued mappings, and characterized realcompactness by means of continuous selections for Tychonoff spaces of non-measurable cardinal. Using their method, we obtain another characterization of realcompactness, but without any cardinal assumption. We also characterize Dieudonné completeness and Lindelöf property in similar formulations.

On a simultaneous selection theorem

Takamitsu Yamauchi (2013)

Studia Mathematica

Valov proved a general version of Arvanitakis's simultaneous selection theorem which is a common generalization of both Michael's selection theorem and Dugundji's extension theorem. We show that Valov's theorem can be extended by applying an argument by means of Pettis integrals due to Repovš, Semenov and Shchepin.

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an F σ -subset A k of X such that d i m A k ≤ k and the restriction f | ( X ∖ A k ) is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about...

On selections of multifunctions

Milan Matejdes (1993)

Mathematica Bohemica

The purpose of this paper is to introduce a definition of cliquishness for multifunctions and to study the search for cliquish, quasi-continuous and Baire measurable selections of compact valued multifunctions. A correction as well as a generalization of the results of [5] are given.

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