Spaces of functions with countably many discontinuities

Haydon, Richard; Moltó, Aníbal; Orihuela, J.

doi:10.1007/s11856-007-0002-1

Cited by 18 publications

(22 citation statements)

References 14 publications

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“…We stress that the previous result for f = id has been used in [28] where it is proved that C p (K) is σ-fragmented when K is a Rosenthal compactum of functions with at most countably many discontinuities.…”

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confidence: 88%

James boundaries and σ-fragmented selectors

2008

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Abstract. 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 S {B(t, ε) : t ∈ T } contains a James boundary for BX * we study different kinds of conditions on T , besides T being countable, which ensure thatWe 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) ∩ J(x) = ∅ for every x ∈ X, then (SP) holds for T = f (X) and in this case X is Asplund; (b) if T is weakly countably K-determined then (SP) holds, X * is weakly countably K-determined and moreover for every James boundary B of BX * we have BX * = co(B)· . Both approaches use Simons' inequality and ideas exploited by Godefroy in the separable case (i.e., when T is countable). While proving (a) we show that X is Asplund if, and only if, the duality mapping has an ε-selector, 0 < ε < 1, that sends separable sets into separable ones. A consequence is that the dual unit ball BX * is norm fragmented if, and only if, it is norm ε-fragmented for some fixed 0 < ε < 1. Our analysis is completed by a characterization of those Banach spaces (not necessarily separable) without copies of 1 via the structure of the boundaries of w * -compact sets of their duals. Several applications and complementary results are proved. Our results extend to the non-separable case results by Godefroy, Contreras-Payá and Rodé.

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confidence: 88%

James boundaries and σ-fragmented selectors

2008

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“…In this way is proved that C(X) is LUR renormable when X is Helly compact. A generalization of this result when X is a particular case of Rosenthal compacts can be found in [10], see also [12].…”

Section: Introductionmentioning

confidence: 82%

Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts

Gul'ko

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Shulikina

et al. 2020

Topology and its Applications

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We show that C(X) admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided X is Fedorchuk compact of spectral height 3. In other words X admits a fully closed map f onto a metric compact Y such that f −1 (y) is metrizable for all y ∈ Y . A continuous map of compacts f : X → Y is said to be fully closed if for any disjoint closed subsets A, B ⊂ X the intersection f (A) ∩ f (B) is finite. For instance the projection of the lexicographic square onto the first factor is fully closed and all its fibers are homeomorphic to the closed interval.

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“…Although the main result was stated for the class CD, it was left unclear in [6] whether this class could coincide with the whole class of Rosenthal compacta. This is not the case, as Pol [9] gave an example of compact space in R \ RK.…”

Section: Introductionmentioning

confidence: 98%