Hereditary invertible linear surjections and splitting problems for selections

Let A+B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h:X→A+B splits into a sum h=f+g of continuous mappings f:X→A and g:X→B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.     

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.     

Spaces determined by selections

A function is a called a weak selection if ψ({x,y})∈{x,y} for every x,y∈X. To each weak selection ψ, one associates a topology τψ, generated by the sets and . Answering a question of S. García-Ferreira and A.H. Tomita [S. García-Ferreira, A.H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (10) (2008) 1105-1110], we show that (X,τψ) is completely regular for every weak selection ψ. We further investigate to what extent the existence of a continuous weak selection on a topological space determines the topology of X. In particular, we answer two questions of V. Gutev and T. Nogura [V. Gutev, T. Nogura, Selection problems for hyperspaces, in: E. Pearl (Ed.), Open Problems in Topology 2, Elsevier B.V., 2007, pp. 161-170].     

Reduction principles in the theory of selections

In the present article many theorems about existence of continuous selections for a set-valued mapping into the metrizable spaces are extended to mappings into non-metrizable spaces.     

Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]²→X$\sigma :{[X]}^2\to X$ such that σ({x,y})∈{x,y}$\sigma (\{x,y\})\in \{x,y\}$ for each {x,y}∈[X]²$\{x,y\}\in {[X]}^2$. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]²${[X]}^2$ and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×Y$X\times Y$ for the following particular cases:

(i)

Both X and Y are spaces with one non-isolated point.     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

Building suitable sets for locally compact groups by means of continuous selections

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.     

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.     

Right inverses of linear maps on convex sets

We study, via continuous selections of multivalued maps, the problem of finding a right inverse to the restriction of a linear map to a convex body.     

Asymmetric norms and optimal distance points in linear spaces

In this paper we analyze the existence of points of a subset S of a linear space X where the shortest distance to a point x of X with respect to an asymmetric norm q is attained (q-nearest points). Since the structure of an asymmetric norm do not provide in general uniqueness of such points—due to the fact that the separation properties in these spaces are in general weaker than in normed spaces—we develop a technique to find particular subsets of the set of q-nearest points—that we call optimal distance points—that are also optimal for the norm qs associated to the asymmetric norm.     

On selections and classes of spaces

Strong paracompactness, Lindelöf number and degree of compactness are characterized in terms of selections of set-valued mappings.     

Fiberwise absolute neighborhood extensors for a class of metrizable spaces

In the paper we study fiberwise absolute neighborhood extensors with respect to some classes of metrizable spaces by means of the local extension properties and the lifting properties of the underlying spaces.     

The hausdorff metric and measurable selections

We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.     

Riemann-measurable selections

LetX be a Polish space equipped with a -finite regular Borel measure . IfE is a metric space andF a set-valued function:X 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.     

Continuous selections avoiding extreme points

It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:X→Y² is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each x∈X, then φ admits a continuous selection f:X→Y such that f(x) is not an extreme point of φ(x) for each x∈X. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521].     

Strong paracompactness and multi-selections

We characterize strong paracompactness in terms of usco multi-selections for closed-valued lower semi-continuous mappings into completely metrizable spaces, thus generalizing recent results obtained by Choban, Mihaylova and Nedev [M. Choban, E. Mihaylova, S. Nedev, On selections and classes of spaces, Topology Appl. 155 (2008) 797-804]. Related results and applications are achieved as well.     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

Moduli of continuity of selections from nonconvex maps

We consider two continuous selection problems related to the differential inclusion F(t, x). Assuming thatF is Hölder or Lipschitz continuous with compact, not necessarily convex values, we provide estimates on the modulus of continuity of these selections.     

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.