Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.     

Hereditarily indecomposable subcontinua of the inverse limit of lexicographic arcs are metric

We show that every hereditarily indecomposable subcontinuum of the inverse limit of copies of the lexicographic arc is metric. It is observed that the technique of proof generalized to the lexicographic cube or hypercubes.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

More remarks on Souslin properties and tree topologies

We investigate separation properties of ω₁-trees. We show that the property γ of Devlin and Shelah is equivalent to hereditary collectionwise normality. We show that monotone normality and divisibility are both equivalent to orderability. Finally we show that Souslin trees are examples of trees with property γ which are not retractable.     

Simply connected homogeneous continua are not separated by arcs

We show that locally connected,simply connected homogeneous continua are not separated by arcs. We ask several questions about homogeneous continua which are inspired by analogous questions in geometric group theory.     

Indecomposability in inverse limits with set-valued functions

In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous functions to be an indecomposable continuum. This condition generalizes and extends those of Ingram and Varagona. Additionally, we demonstrate a method of constructing upper semi-continuous functions whose inverse limit has the full projection property.     

Inverse limits of arcs and of simple closed curves with confluent bonding mappings

It is proved that Knaster's type continua and solenoids can be considered as inverse limits of arcs and of circles with confluent bonding mappings. Several other classes of bonding mappings, which are relative to confluent ones, also are discussed.     

A non-metric perfectly normal hereditarily indecomposable continuum

We construct an example of a non-metric perfectly normal hereditarily indecomposable continuum. The example is constructed as an inverse limit of non-metric analogues of solenoids. Theorems needed to insure perfect normality are stated and proven. It is shown that the example cannot be embedded in a countable product of Hausdorff arcs.     

Limits of inverse limits

We study the following problem: if a sequence of graphs of upper semi-continuous set valued functions fn converges to the graph of a function f, is it true that the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f?     

Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts

We characterize metric spaces X whose hyperspaces X² (or Bd(X)) of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.     

Continua for which connected partitions are compact

We investigate connected partitions of continua into compacta. In particular, we consider continua with property that every connected partition into compacta is compact. We characterize graphs which have this property as the trees and the simple closed curve. Dendrites are shown to have the property. An example of a nonlocally connected continuum with the property is also given.     

Paths through inverse limits

In Bani?, ?repnjak, Merhar and Milutinovi? (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions fn:X→X² converges to the graph of a continuous single-valued function f:X→X, then the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f. In this paper a more general result is presented in which surjectivity of fn is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.     

A characterization of inverse limits of Q-bundle maps

In this paper we prove that a surjection between metric compacta is a hereditary shape equivalence if and only if it is an inverse limit of trivial Q-bundle maps. This result was conjectured by T.B. Rushing. Near-homeomorphisms are instrumental to the proof.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Fréchetness of inverse limits and products

Let X be a Suslin-Borel set in a compact space. It is proved that X is either σ-scattered or contains a compact perfect set. If X is first countable, the result remains valid when X is a Suslin-Borel set in a Prohorov space. It is also proved that every first countable Prohorov space is a Baire space.     

Countable connected Hausdorff and Urysohn bunches of arcs in the plane

In this paper, we answer a question by Krasinkiewicz, Reńska and Sobolewski by constructing countable connected Hausdorff and Urysohn spaces as quotient spaces of bunches of arcs in the plane. We also consider a generalization of graphs by allowing vertices to be continua and replacing edges by not necessarily connected sets. We require only that two “vertices” be in the same quasi-component of the “edge” that contains them. We observe that if a graph G cannot be embedded in the plane, then any generalized graph modeled on G is not embeddable in the plane. As a corollary we obtain not planar bunches of arcs with their natural quotients Hausdorff or Urysohn. This answers another question by Krasinkiewicz, Reńska and Sobolewski.     

Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.     

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Let T be a tent map with the slope strictly between and 2. Suppose that the critical point of T is not recurrent. Let K denote the inverse limit space obtained by using T repeatedly as the bonding map. We prove that every homeomorphism of K to itself is isotopic to some power of the natural shift homeomorphism.     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.