Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Bazylevych, Lidia; Repovš, Dušan; Zarichnyi, Mykhailo

doi:10.1016/j.topol.2007.02.014

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…The required statement follows from Theorem 4.1 in [14] combined with the results of [15], which provide an analogous statement for the functors cc and P.…”

Section: Proposition 3 the Space Os X ( ) Is An Absolute Retract If mentioning

confidence: 80%

“…Proposition 3.1 in [14] and the openness of the functor of probability measures yield the following statement: It was shown in [5] that the monad O generated by the functor of weakly additive functionals weakly preserves preimages (Theorem 4.2). Since OH and OS are submonads of O, they also weakly preserve preimages.…”

Section: Proposition 1 Let F : X → Y Be An Arbitrary Surjective Mappmentioning

confidence: 99%

“…It follows from the remarks on OS made above that O S is, in fact, isomorphic to the composition of the functors cc and P. Some properties of the functor cc were studied in [14]. For an arbitrary convex compactum X, ccX is defined as the set of all nonempty, closed, convex subsets of X and is considered as a subspace of exp X .…”

Section: Proposition 1 Let F : X → Y Be An Arbitrary Surjective Mappmentioning

confidence: 99%

“…The following statement can be obtained by applying the same arguments as in [ From the remarks on OS made in the first section one can see that OS is in fact isomorphic to the composition of the functors cc and P . Some properties of the functor cc were studied in [8]. For any convex compact X, ccX is defined to be the set of all nonempty closed convex subsets of X, ccX is considered as the subspace of exp X.…”

Section: Definitions and Factsmentioning

confidence: 99%

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On geometric properties of functors of positive-homogenous and semiadditive functionals

Karchevska

2011

Ukr Math J

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515.12We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH X ( ) is an absolute retract if and only if X is an open-generated compactum, and OS X ( ) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω 1 . We investigate the softness of mappings of multiplication of monads generated by these functors. IntroductionIn [1], Fedorchuk posed the following general question: How do functors affect certain geometric properties of spaces and mappings between them? Geometric properties are understood here as the property of a space to be an absolute retract, the property of a mapping to be soft or to be a Tikhonov fibering, etc.There have been many investigations in this direction involving such functors as the hyperspace functor, probability-measure functor, superextension functor, and other functors (see, e.g., [2]).As an example, consider the probability-measure functor and the superextension functor. There is a natural structure of linear convexity on the space P X ( ) for an arbitrary compactum X. For λ, de Groot [3] constructed an abstract (not linear) convexity on an arbitrary space of the form λ( ) X , and this convexity is binary, whereas the linear convexity on P X ( ) is not binary. The functors P and λ have different geometric properties. For example, the functor P, in contrast to λ, does not give absolute retracts in weights greater than ω 1 .The notion of convexity considered in the present paper is considerably broader than the classic one because it does not depend on the presence of a linear structure on a space. Our approach is based on the notion of topological convexity introduced in [4]. In [5], Radul associated every monad F with a certain abstract convexity structure on every space FX, where F is the functorial part of the monad F. If a monad F is an L-monad that weakly preserves preimages, then this convexity generates a topology of the space FX. It was also shown in [5] that L-monads that weakly preserve preimages and for which the introduced convexity structure is binary possess good geometric properties in weights greater than ω 1 (Theorem 3.3).In the present paper, we consider functors OS and OH (introduced in [6, 7]) that generate L-monads. The monad OS does not generate binary convexities, whereas the monad OH is binary, which is one of the factors that cause the difference in the geometric properties of these monads: the properties of OS are close to those of P, and the properties of OH are closer to those of λ. Definitions and FactsIn the present paper, we consider only objects and morphisms of the category Comp, i.e., compact Hausdorff spaces (or, briefly, compacta) and continuous mappings between them.

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“…The required statement follows from Theorem 4.1 in [14] combined with the results of [15], which provide an analogous statement for the functors cc and P.…”

Section: Proposition 3 the Space Os X ( ) Is An Absolute Retract If mentioning

confidence: 80%

Section: Proposition 1 Let F : X → Y Be An Arbitrary Surjective Mappmentioning

confidence: 99%

Section: Proposition 1 Let F : X → Y Be An Arbitrary Surjective Mappmentioning

confidence: 99%

Section: Definitions and Factsmentioning

confidence: 99%

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On geometric properties of functors of positive-homogenous and semiadditive functionals

Karchevska

2011

Ukr Math J

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“…Given a nonempty compact subset V of a locally convex space W , denote by h(V ) the closed convex hull of V ; i.e., define a continuous mapping h : exp c (V ) → cc(W ) [20].…”

Section: Sublinear Operators and Multivalued Mappingsmentioning

confidence: 99%

Extension conditions for bounded linear and sublinear operators with values in lindenstrauss spaces

Linke

2010

Sib Math J

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Dimension of Extremal Boundary of the Space of Semiadditive Functionals

Djabbarov

2020

J Math Sci

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Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Cited by 6 publications

References 13 publications

On geometric properties of functors of positive-homogenous and semiadditive functionals

On geometric properties of functors of positive-homogenous and semiadditive functionals

Extension conditions for bounded linear and sublinear operators with values in lindenstrauss spaces

Dimension of Extremal Boundary of the Space of Semiadditive Functionals

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