Valentin Gutev

Vietoris continuous selections and disconnectedness-like properties

Suppose that X is a Hausdorff space such that its Vietoris hyperspace (F(X), τV ) has a continuous selection. Do disconnectedness-like properties of X depend on the variety of continuous selections for (F(X), τV ) and vice versa? In general, the answer is “yes” and, in some particular situations, we were also able to set proper characterizations.

Selection theorems under an assumption weaker than lower semi-continuity

Abstract Every set-valued mapping satisfying an assumption weaker than lower semi-continuity admits a lower semi-continuous selection. Besides in the selection theory, this result is also succesful in solving the problem of extending lower semi-continuous mappings from arbitrary to G δ -subsets of metric spaces.

Extreme selections for hyperspaces of topological spaces

Abstract We study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F (X) of closed non-empty subsets. Involving extreme selections for F (X) , we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F (X) , we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point.

Continuous selections and -spaces

A characterization of paracompact C-spaces via continuous selections avoiding Z∞-sets is given. The result is applied to prove a countable sum theorem for paracompact C-spaces, and to obtain a new partial solution of a question raised by E. Michael.

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Zn-sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrands theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michaels result.

Selection problems for hyperspaces

Publisher Summary This chapter discusses selection problems for hyperspaces. For a T 1 -space X , let ℱ( X ) be the set of all nonempty closed subsets of X. ℱ( X ) is endowed with the Vietoris topology TV , and called the Vietoris hyperspace of X. A space X is orderable (or linearly orderable) if the topology of X coincides with the open interval topology on X generated by a linear ordering on X . A space X is sub-orderable (or generalized ordered) if it can be embedded into an orderable space. A space X is weakly orderable if there exists a coarser orderable topology on X . In all these cases, the corresponding linear order on X is called compatible for the topology of X or a compatible order for X . A selection f : ℱ 2 ( X ) → X is usually called a weak selection for X . This chapter elaborates about weak selections and properties that follow from orderability, and it discusses the concepts of topological well-ordering and selections, in detail. A discussion on selections and disconnectedness-like properties is also presented in the chapter.

Fell continuous selections and topologically well-orderable spaces

The present paper improves a result of [3] by showing that a space X is topologically well-orderable if and only if there exists a selection for F2(X) which is continuous with respect to the Fell topology on F2(X). In particular, this implies that F(X) has a Fell-continuous selection if and only if F2(X) has a Fell-continuous selection.

Cantor set selectors

Abstract If every l.s.c. mapping from the Cantor set C to the closed subsets of a metric space X admits a u.s.c. selection, then X is a Baire space and either X is scattered or X contains a copy of C .

Selections and Hyperspaces

This survey deals with some historical background and recent developments in the area of selections for hyperspace topologies. It was written with the hope that the presented material may pique an interest in the hyperspace selection problem—a problem with a fascinating history and appealing applications.

Baire spaces and Vietoris hyperspaces

We prove that if the Vietoris hyperspace CL(X) of all nonempty closed subsets of a space X is Baire, then all finite powers of X must be Baire spaces. In particular, there exists a metrizable Baire space X whose Vietoris hyperspace CL(X) is not Baire. This settles an open problem of R. A. McCoy stated in 1975.