Filament sets and homogeneous continua

New tools are introduced for the study of homogeneous continua. The subcontinua of a given continuum are classified into three types: filament, non-filament, and ample, with ample being a subcategory of non-filament. The richness of the collection of ample subcontinua of a homogeneous continuum reflects where the space lies in the gradation from being locally connected at one extreme to indecomposable at another. Applications are given to the general theory of homogeneous continua and their hyperspaces.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Arc-smoothness in hyperspaces

Let X be a metric continuum and 2^x (C(X)) denote the hyperspace of closed subsets (subcontinua) of X. The concept of arc-smoothness, which is a special type of contractibility, is investigated in 2^x and C(X). Results are obtained about hyperspaces of locally connected continua, about continua for which C(X) and the cone over X are homeomorphic, about Whitney levels in C(X), and about hyperspaces of hereditarily indecomposable continua. Some examples are given and several natural questions are raised.     

The surjective semispan for Hausdorff continua

It is well known that Tychonoff spaces are those whose topology is induced by a uniformity. We use this fact to give two characterizations of chainable continua; the first one in terms of V-chains and the other one in terms of V-maps. We also define the surjective semispan for Hausdorff continua and we prove that chainable continua has empty surjective semispan. As a consequence of this result we obtain that each map from a continuum onto a chainable continuum is universal; in particular, chainable continua have the fixed point property.     

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.     

Remainders of arcwise connected compactifications of the plane

A. Lelek asked which continua are remainders of locally connected compactifications of the plane. In this paper we study a similar problem with local connectedness replaced by arcwise connectedness. (Each locally connected continuum is arcwise connected.) We give the following characterization: a continuum X is pointed 1-movable if and only if there is an arcwise connected compactification of the plane with X as the remainder.     

Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

Hereditarily indecomposable subcontinua of inverse limits of Souslin arcs are metric

We prove that if Si is a Souslin arc (a Hausdorff arc that is the compactification of a Souslin line) for each i and , then every hereditarily indecomposable subcontinuum of X is metric. Since every non-degenerate hereditarily indecomposable continuum that is an inverse limit on metric arcs is a pseudo-arc, it follows that such an X would be a pseudo-arc or a point.     

Pseudo-path connected homogeneous continua

The main result of this paper states that every homogeneous pseudo-path connected continuum is weakly chainable, or equivalently, every homogeneous continuum connected by continuous images of the pseudo-arc is itself a continuous image of the pseudo-arc. We notice that even though there exist homogeneous path connected continua that are not continuous images of an arc (Prajs, 2002), they all are continuous images of the pseudo-arc.     

Indecomposable continua and the Julia sets of polynomials, II

We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree d?2 to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors.We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable.     

The Sarkovskii order for periodic continua

Suppose f is a map of a continuum X onto itself. A periodic continuum of f is a subcontinuum K of X such that fn[K]=K for some positive integer n. A proper periodic continuum of f is a periodic continuum of f that is a proper subcontinuum of X. A proper periodic continuum of f is maximal if and only if X is the only periodic continuum that properly contains it. In this paper it is shown that the maximal proper periodic continua of a map of a hereditarily decomposable chainable continuum onto itself follow the Sarkovskii order, provided the maximal proper periodic continua are disjoint. The case in which the Sarkovskii order does not hold reduces to the scenario in which the map's domain is the union of two overlapping period-two continua, each of which is maximal.     

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.     

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.     

On continuously irreducible continua

We study continuously irreducible continua and characterize them as those continua of type λ for which the set function T is continuous. Using results by Mohler and Oversteegen, we present a new family of one-dimensional continua for which the set function T is continuous and no element of the family contains a pseudo-arc. We study the hyperspaces of these continua.     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.     

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.

     

Rigid continua and transfinite inductive dimension

We introduce a general method of resolving first countable, compact spaces that allows accurate estimate of inductive dimensions. We apply this method to construct, inter alia, for each ordinal number α>1 of cardinality ?c, a rigid, first countable, non-metrizable continuum Sα with . Sα is the increment in some compactification of [0,1) and admits a fully closed, ring-like map onto a metric continuum. Moreover, every subcontinuum of Sα is separable. Additionally, Sα can be constructed so as to be: (1) a hereditarily indecomposable Anderson-Choquet continuum with covering dimension a given natural number n, provided α>n, (2) a hereditarily decomposable and chainable weak Cook continuum, (3) a hereditarily decomposable and chainable Cook continuum, provided α is countable, (4) a hereditarily indecomposable Cook continuum with covering dimension one, or (5) a Cook continuum with covering dimension two, provided α>2.We also produce a chainable and hereditarily decomposable space Sω(c⁺) with , , trind₀Sω(c⁺) and trInd₀Sω(c⁺) all equal to ω(c⁺), the first ordinal of cardinality c⁺.     

Ultra smooth continua are smooth

In this paper we to define and study a class of metric continua called ultra smooth, which (by the definition) form a subclass of the class of continua being hereditarily unicoherent at some point. The major result is that every ultra smooth continuum is smooth.     

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.