σ-Fragmentability of Multivalued Maps and Selection Theorems

Jayne, J. E.; Orihuela, J.; Pallarés, A.J.; Vera, G.

doi:10.1006/jfan.1993.1127

Cited by 45 publications

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“…[18] Let (X, τ ) be a topological space and (Y, µ) a uniform space. We say that a function f : X → Y is fragmented if for every nonempty subset A of X and every entourage ε ∈ µ there exists an open subset O of X such that O ∩ A is nonempty and the set f (O ∩ A) is ε-small in Y .…”

Section: Fragmented Mapsmentioning

confidence: 99%

A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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Abstract. We show that for arbitrary linearly ordered set (X, ≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y, d) the compact space BVr (X, Y ) of all functions X → Y with variation ≤ r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M + (X, Y ) of all order preserving maps X → Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.

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Section: Fragmented Mapsmentioning

confidence: 99%

A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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“…Note that in Definition 6.1.1 when Y = X, f = id X and µ is a metric uniform structure, we get the usual definition of fragmentability [33]. For the case of functions see also [32].…”

Section: Fragmented Maps and Familiesmentioning

confidence: 99%

Hereditarily non-sensitive dynamical systems and linear representations

Glasner¹,

Megrelishvili

2006

Colloq. Math.

117

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Abstract. For an arbitrary topological group G any compact G-dynamical system (G, X) can be linearly G-represented as a weak * -compact subset of a dual Banach space V * . As was shown in [45] the Banach space V can be chosen to be reflexive iff the metric system (G, X) is weakly almost periodic (WAP). In this paper we study the wider class of compact G-systems which can be linearly represented as a weak * -compact subset of a dual Banach space with the Radon-Nikodým property. We call such a system a Radon-Nikodým system (RN). One of our main results is to show that for metrizable compact G-systems the three classes: RN, HNS (hereditarily not sensitive) and HAE (hereditarily almost equicontinuous) coincide. We investigate these classes and their relation to previously studied classes of G-systems such as WAP and LE (locally equicontinuous). We show that the Glasner-Weiss examples of recurrent-transitive locally equicontinuous but not weakly almost periodic cascades are actually RN. Using fragmentability and Namioka's theorem we give an enveloping semigroup characterization of HNS systems and show that the enveloping semigroup E(X) of a compact metrizable HNS G-system is a separable Rosenthal compact, hence of cardinality ≤ 2 ℵ 0 . We investigate a dynamical version of the Bourgain-Fremlin-Talagrand dichotomy and a dynamical version of Todorcević dichotomy concerning Rosenthal compacts.

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“…First, σ-slicely continuous maps introduced in [18] provide a characterisation of the existence of an equivalent LUR norm in a Banach space. On the other hand, σ-fragmentable maps were introduced in [15] in order to study selection problems. By D U (C, M ) we denote the set of maps from D(C, M ) which are moreover uniformly continuous on bounded subsets of C. That technical condition is necessary in order to perform several operations motivated by the geometrical study of the RNP property, which ensures nice properties for this class of maps.…”

Section: Introductionmentioning

confidence: 99%

Maps with the Radon–Nikodým Property

García-Lirola

Raja

2017

Set-Valued Var. Anal

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Abstract. We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikodým property to a "less linear" frame. We note that a certain part of the theory can be developed in rather great generality. Indeed, we establish that the elements of the dual which are "strongly slicing" for a given uniformly continuous dentable function form a dense G δ subset of the dual. As a consequence, the space of uniformly continuous dentable maps from a closed convex bounded set to a Banach space is a Banach space. However some interesting applications, as Stegall's variational principle, are no longer true beyond the usual hypotheses, sending us back to the classical case. Moreover, we study the connection between dentability and approximation by delta-convex functions for uniformly continuous functions. Finally, we show that the dentability of a set is closely related with the dentability of delta-convex maps defined on it.

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σ-Fragmentability of Multivalued Maps and Selection Theorems

Cited by 45 publications

References 0 publications

A note on tameness of families having bounded variation

A note on tameness of families having bounded variation

Hereditarily non-sensitive dynamical systems and linear representations

Maps with the Radon–Nikodým Property

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