Modified continuity and a generalisation of Michael's selection theorem

We modify the definitions of continuity and lower semicontinuity for single-valued mappings and upper and lower semicontinuity for set-valued mappings. For single-valued mappings we have a generalisation of Osgood's theorem and for set-valued mappings we have an extension of Fort's theorem and a generalisation of Michael's selection theorem producing a densely defined selection with a natural continuity property relative to the domain.     

On connections between some selection and approximation results for multivalued maps

We study the general conditions on a family of values of a multivalued map under which the existence of single-valued approximations may be derived from continuous selection results.     

Selections for quasi-l.s.c. mappings with uniformly equi-LCn range

Every quasi-lower semi-continuous (q.l.s.c.) mapping admits a lower semi-continuous (l.s.c.) selection preserving all important (from the selection point of view) properties of the former mapping. Special-type extensions of l.s.c. mappings are established on this base.     

Generalized selection theorems without convexity

In this paper, we extend new selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X to general nonconvex settings. On the basis of the Kim-Lee theorem and the Horvath selection theorem, we first show that any a.l.s.c. C-valued multifunction admits a continuous selection under a mild condition of a one-point extension property. Finally, we apply a fundamental selection theorem, due to Ben-El-Mechaiekh and Oudadess, to modify our selection theorems by adjusting a closed subset Z of X with its covering dimension dim_XZ≤0. The results derived here generalize and unify various earlier ones from classic continuous selection theory.     

Continuous selections of multifunctions with weakly convex values

We obtain some selection theorems for multifunctions with weakly convex values. For this purpose, some new properties of weakly convex sets in a Hilbert space are investigated. We also present some examples showing the importance of various assumptions in these selection theorems.     

Modifications of sequence selection principles

We study modifications of the sequence selection principles for real functions obtained by allowing other than continuous functions and/or by replacing the pointwise convergence by quasi-normal or discrete convergence. We show that for important families of real functions we obtain already known sequence selection principles. Replacing the pointwise convergence we obtain plenty of sequence selection principles and we try to find equivalences or implications among them. Finally we attempt to show that these modifications are useful tools for proofs of known results or their strengthening.     

Selections and totally disconnected spaces

We demonstrate that for every n<ω there exists a separable completely metrizable space Xn which has a continuous selection for its Vietoris hyperspace of nonempty compact subsets, but dim(Xn)=n. Related results and open problems are discussed.     

Selections and topological convexity

A simple natural proof of van de Vel's selection theorem for topological convex structures is given. The technique developed to achieve this proof allows to give also a direct simple proof of the classical Michael's selection theorem in Fréchet spaces, and the Horvath's selection theorem in metric l.c.-spaces.     

Moscow spaces and selection theory

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We present here the selection version of this result and also (by means of two different approaches) we use selection theory to characterize the open bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

Extensions by Means of Expansions and Selections

We provide proper mapping-characterizations of some embedding-like properties weaker than -embedding. For instance, we show that a subset A of a space X is -embedded in X if and only if for every continuous map g: A → Y into a Banach space Y of weight w(Y) ⩽ λ, there exists a continuous set-valued mapping φ of X into the nonempty compact subsets of Y such that g is a selection for φ∣A (i.e., g(x) ∈ φ(x) for every x ∈ A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of -embedding which makes more transparent the relation ‘’. Other weak components of -embedding are described in terms of expansions and selections, possible applications are demonstrated as well.     

L'espace des sélections d'une dendrite

We prove that the selection space of a nondegenerate dendrite is homeomorphic to ?².     

On closedness assumptions in selection theorems

We begin by a short survey of various attempts in selection theory to avoid the closedness assumption for values of multivalued mappings. We collect special cases when Michael's Gδ-problem admits an affirmative solution and we prove some unified theorems of such type. We also show that in general this problem has a negative solution. In comparison with a recent result of Filippov, we work directly in the Hilbert cube rather than in the space of all probabilistic measures endowed with different topologies.     

On two selection principles and the corresponding games

We give some further results on two selection principles involving dense families of open sets and the corresponding games.     

Multivalued maps, selections and dynamical systems

Under suitable hypotheses the well known notion of first prolongational set J⁺ gives rise to a multivalued map which is continuous when the upper semifinite topology is considered in the hyperspace of X. Some important dynamical concepts such as stability or attraction can be easily characterized in terms of ψ and moreover, the classical result that an attractor in Rn has the shape of a finite polyhedron can be reinforced under the hypotheses that the mapping ψ is small and has a selection.     

Weak orderability of topological spaces

In 1951 Ernest Michael wrote a definitive seminal article on hyperspaces [E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951) 152-182] raising a general question that became known as Michael's selection problem for hyperspaces. The present paper contains a detailed discussion on particular aspects of this problem, also some further open questions.     

Selections for paraconvex-valued mappings on non-paracompact domains

We prove that Michael?s paraconvex-valued selection theorem for paracompact spaces remains true for C^′(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.     

Extending paired multifunctions

As a rule, the classical Michael-type selection theorems for the existence of single-valued selections are analogues and, in certain respects, generalisations of ordinary extension theorems. In contrast to this, the theorems for the existence of multi-selections deal with natural generalisations of cover properties of topological spaces. This paper continues the study of the latter problem, and its main purpose is to furnish a mapping characterisation of a cover-extension property—the so-called Katětov spaces.     

Collectionwise normality and selections into Hilbert spaces

A T₁-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbert space with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev and S. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1-2) (1997) 87-95].     

Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces

The concept of lower semicontinuity is extended to functions mapping into partially ordered spaces. A study is made of spaces of such lower semicontinuous functions under the epi-topology. These spaces are subspaces of hyperspaces with the Fell topology. The closure of such a function space in the hyperspace is characterized for certain spaces. A continuous selection theorem is established, showing that most such function spaces are not ech-complete.