Scorza Dragoni type theorems
Searching for more absolute CR-epic spaces.
Seeking a network characterization of Corson compacta
We say that a collection of subsets of has property if there is a set and point-countable collections of closed subsets of such that for any there is a finite subcollection of such that . Then we prove that any compact space is Corson if and only if it has a point-- base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by...
Selected results on measurable selections
Selection from Upper Semi-Continuous Compact-Valued Mappings.
Selection principles and Baire spaces
Selection principles and upper semicontinuous functions
In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and in terms of upper semicontinuous functions
Selection Theorem for Systems with Inheritance
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural”...
Selection theorem in L¹
Let F be a multifunction from a metric space X into L¹, and B a subset of X. We give sufficient conditions for the existence of a measurable selector of F which is continuous at every point of B. Among other assumptions, we require the decomposability of F(x) for x ∈ B.
Selection theorems for partitions of Polish spaces
Selections and approaching points in products
The present paper aims to furnish simple proofs of some recent results about selections on product spaces obtained by García-Ferreira, Miyazaki and Nogura. The topic is discussed in the framework of a result of Katětov about complete normality of products. Also, some applications for products with a countably compact factor are demonstrated as well.
Selections and near-selections in metric linear spaces without local convexity
We characterize the AR property in convex subsets of metric linear spaces in terms of certain near-selections.
Selections and representations of multifunctions in paracompact spaces
Let (X,T) be a paracompact space, Y a complete metric space, a lower semicontinuous multifunction with nonempty closed values. We prove that if is a (stronger than T) topology on X satisfying a compatibility property, then F admits a -continuous selection. If Y is separable, then there exists a sequence of -continuous selections such that for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.
Selections and suborderability
We extend van Mill-Wattel's results and show that each countably compact completely regular space with a continuous selection on couples is suborderable. The result extends also to pseudocompact spaces if they are either scattered, first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. A zero-selection is a selection on the hyperspace of closed sets which chooses always an isolated point of a set. Extending Fujii-Nogura...
Selections and weak orderability
We answer a question of van Mill and Wattel by showing that there is a separable locally compact space which admits a continuous weak selection but is not weakly orderable. Furthermore, we show that a separable space which admits a continuous weak selection can be covered by two weakly orderable spaces. Finally, we give a partial answer to a question of Gutev and Nogura by showing that a separable space which admits a continuous weak selection admits a continuous selection for all finite sets.
Sélections continues
Selections generating new topologies.
Every (continuous) selection for the non-empty 2-point subsets of a space X naturally defines an interval-like topology on X. In the present paper, we demonstrate that, for a second-countable zero-dimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7].
Selections of set-valued stochastic processes.
Selections on -spaces
We show that if is an uncountable AD (almost disjoint) family of subsets of then the space does not admit a continuous selection; moreover, if is maximal then does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.
Selections that characterize topological completeness
We show that the assertions of some fundamental selection theorems for lower-semicontinuous maps with completely metrizable range and metrizable domain actually characterize topological completeness of the target space. We also show that certain natural restrictions on the class of the domains change this situation. The results provide in particular answers to questions asked by Engelking, Heath and Michael [3] and Gutev, Nedev, Pelant and Valov [5].