Extension theorems for some classes of continua

It is proved that every mapping from a proper subcontinuum of a hereditarily unicoherent continuum X onto the Knaster's indecomposable continuum (onto a cone over a zerodimensional compact metric set) can be extended to a mapping defined on X.Similarly, every mapping from a proper subcontinuum of a hereditarily indecomposable continuum onto a pseudoarc can be extended to a mapping defined on the whole space.Both of the above results are generalizations of the author's earlier results to the nonmetric case. As a consequence it is obtained that a pseudoarc is continuously n-homogeneous.     

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.     

Local homeomorphisms onto nonunicoherent continua

We show that ifY is a nonunicoherent continuum and if 1 n < , then there is a local homeomorphism of degreen from some continuumX ontoY.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

On selections and classes of spaces

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

Finite linear coverings of locally connected continua

A concept of finite coverings of continua with a linear order of their members is given. A characterization is obtained of hereditarily locally connected continua which have a finite supremum of cardinalities of the considered coverings.     

Mappings between inverse limits of continua with multivalued bonding functions

We investigate the limit mappings between inverse limits of continua with upper semi-continuous bonding functions. Results are obtained when the coordinate mappings are surjective, one-to-one or homeomorphisms. We construct examples showing the hypothesis of the theorems are essential. Further, we construct an example showing that, unlike for the inverse limits with single valued maps, properties of being monotone, confluent or weakly confluent mappings between factor spaces are not preserved in the inverse limit map.     

A method for constructing ordered continua

We present a method for constructing ordered continua. We illustrate our method by constructing (i) a new order-homogeneous non-reversible continuum, and (ii) an ordered continuum with a minimal set of continuous self-maps.     

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.     

Local planarity in one-dimensional continua

In this paper we address a problem posed by W. Lewis at the Second International Conference on Continuum Theory held at BUAP, Puebla, Mexico. Lewis asked for a characterization of local-planarity in inverse limit spaces of finite graphs in terms of the dynamics of the bonding maps. We give some sufficiency conditions and show that points at which our sufficiency conditions do not guarantee the space is locally planar, the problem requires a solution to the harder problem of characterizing planarity in inverse limits of graphs. We also examine the case of an inverse limit generated by a single map, f, on a single graph, G. Assuming that f has finitely many turning points and is non-contracting, we characterize local planarity in terms of the dynamics of f.     

On some classes of Lindelöf Σ-spaces

We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ₁ is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.     

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.     

Omega-limit sets in hereditarily locally connected continua

In this paper we give a topological characterization of ω-limit sets in hereditarily locally connected continua. Moreover, we characterize also orbit-enclosing ω-limit sets in these spaces.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

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.     

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 the space of subgroups of a compact group I

A theory of the family of closed subgroups of a compact topological group is developed, using the topological notion of a hyperspace. Basic properties of this “space of subgroups” are explored.     

On the space of subgroups of a compact group II

Let G be a compact group. Let S(G), C(G), N(G) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology.Then: (1) S(G) and C(G) are never connected; (2) N(G) is always totally disconnected; (3) C(G) is totally disconnected if and only if G is totally disconnected; and (4) S(G) is totally disconnected if and only if G/Z(G) is totally disconnected.Further: for totally disconnected G (equivalently, profinite G) (5) S(G), C(G) and N(G) are κ-metrisable; (6) S(G), C(G) and N(G) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].