Circle-like Continua Research Articles

Characterizing Inverse Sequences For Which Their Inverse Limits Are Homeomorphic

In [11], Mioduszewski characterized inverse sequences of polyhedrafor which their inverse limits are homeomorphic. In this article, we obtain amore general characterization: we characterize inverse sequences of arbitrary compactmetric spaces and continuous single-valued functions for which their inverselimits are homeomorphic. In our approach, set-valued functions are used insteadof continuous single-valued functions in almost commutative diagrams. Usingthis characterization we give an alternative proof that the Brouwer-Janiszewski-Knaster continuum and the pseudo-arc are circle-like continua.

Pseudo-solenoids are not continuously homogeneous

The topological space X is a pseudo-solenoid if X is a hereditarily indecomposable, non-chainable, and circle-like continuum. It is shown that no such continuum is continuously homogeneous. Specifically, there are two points in X, identified as z and 1¯, such that for any continuous surjection f:X→X, f(1¯)≠z. This answers a question of J.J. Charatonik in the negative.

On indecomposable [formula omitted]-homogeneous circle-like continua

We exhibit an indecomposable 12-homogeneous circle-like continuum, answering a question of Neumann-Lara, Pellicer-Covarrubias and Puga-Espinosa from 2005. Our solenoidal construction is an extension of Jim Rogersʼ proof of homogeneity of solenoids of pseudoarcs and leads in turn to an uncountable family of mutually nonequivalent examples. In addition we prove that for an indecomposable 12-homogeneous continuum Y the two orbits of Homeo(Y) are uncountable.

Half-homogeneous indecomposable circle-like continuum

A continuum is said to be 12-homogeneous if there are exactly two types of points. We give an example of a 12-homogeneous indecomposable circle-like continuum. This answers a question of V. Neumann-Lara, P. Pellicer-Covarrubias and I. Puga.

Characterizing chainable, tree-like, and circle-like continua

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\mathcal U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\mathcal U_4$-map $f\colon X\to Y$ onto a tree (resp. onto the circle, onto the interval). A

The solenoids are the only circle-like continua that admit expansive homeomorphisms

A homeomorphism $h:X\rightarrow X$ of a compactum $X$ is <i>expansive</i> provided that for some fixed $c>0$ and any distinct $x, y\in X$ there exists an integer $n$, dependent only on $x$ and $y$, such that ${d}(h^n(x),h^n(y))>c$. It is shown that if $

A brief historical view of continuum theory

We explore a few topics in continuum theory from their roots. Specifically, we examine the evolution of the definition of continuum and then restrict most of our attention to one-dimensional continua. Particular attention is paid to indecomposable continua, the fixed point property, hereditary equivalent continua, homogeneous continua, chainable continua and span of continua. In this paper, we give an inverse limit description of an indecomposable circle-like continuum that is homeomorphic to the first example of an indecomposable continuum given by L.E.J. Brouwer in 1910.

Local connectivity functions on arcwise connected spaces and certain continua

Types of spaces are given on which every local connectivity function is a connectivity function, a connected function, or a Darboux function. A complete determination such spaces is obtained when the spaces are assumed to be arc-like continua or circle-like continua. Results provide answers to a question asked by Stallings.

The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua

A homeomorphism f: X -* X of a compactum X with metric d is expansive if there is c > 0 such that if x, y E X and x 7& y, then there is an integer n E Z such that d(f n(x), fn (y)) > c. It is well-known that padic solenoids Sp (p > 2) admit expansive homeomorphisms, each Sp is an indecomposable continuum, and Sp cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each 1 < n < 3, does there exist a plane continuum X so that X admits an expansive homeomorphism and X separates the plane into n components? For the case n = 2, the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if f: X -* X is an expansive homeomorphism of a circlelike continuum X, then f is itself weakly chaotic in the sense of Devaney.

Shape Equivalences of Whitney Continua of Curves

By a compactum, we mean a compact metric space. A continuum is a connected compactum. A curve is a 1-dimensional continuum. Let X be a continuum and let C(X) be the hyperspace of (nonempty) subcontinua of X, C(X) is metrized with the Hausdorff metric (e.g., see [12] or [18]). One of the most convenient tools in order to study the structure of C(X) is a monotone map ω:C(X) → [0, ω(X)] defined by H. Whitney [25]. A map ω:C(X) → [0, ω(X)] is said to be a Whitney map for C(X) provided thatThe continua {ω−1} (0 &lt; t &lt; ω(X)) are called the Whitney continua of X. We may think of the map ω as measuring the size of a continuum. Note that ω−1(0) is homeomorphic to X and ω−1(ω(X)) = {X}. Naturally, we are interested in the structures of ω−1(t)(0 &lt; t &lt; ω(X)). In [14], J. Krasinkiewicz proved that if X is a circle-like continuum and ω is any Whitney map for C(X), then for any 0 &lt; t &lt; ω(X)ω−1(t) is shape equivalent to X, i.e., Sh ω−1(t) = Sh X (e.g., see [1] or [17]). In [8], we proved the following: If one of the conditions (i) and (ii) is satisfied, then the shape morphismwhich is defined in [7] and [8], is a shape equivalence.

Some Properties of Hyperspaces with Applications to Continua Theory

In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainableIn [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

A Cohomological Characterization of Preimages of Nonplanar, Circle-like Continua

A Cohomological Characterization of Preimages of Nonplanar, Circle-like Continua

A cohomological characterization of preimages of nonplanar, circle-like continua

Let G G be an additively written Abelian group, and let P = { p 1 , p 2 , p 3 , ⋯ } P = \{ {p_1},{p_2},{p_3}, \cdots \} be a sequence of positive integers. An element g g in G G is said to have infinite P P -height if (1) g ≠ 0 g \ne 0 , (2) each p i &gt; 1 {p_i} &gt; 1 , and (3) for each positive integer n n , there is an element h h in G G such that ( p 1 p 2 ⋯ p n ) h = g ({p_1}{p_2} \cdots {p_n})h = g . The purpose of this paper is to prove the following Theorem. If X X is a continuum, then the following are equivalent: (1) H 1 ( X ) {H^1}(X) contains an element of infinite P P -height, for some sequence P P of positive integers; (2) X X can be mapped onto a solenoid; (3) X X can be mapped onto a nonplanar, circle-like continuum. Here H 1 ( X ) {H^1}(X) is Alexander-Čech cohomology with integral coefficients.

On the hyperspaces of snake-like and circle-like continua

On the hyperspaces of snake-like and circle-like continua

Continua with cones homeomorphic to hyperspaces

We investigate continua with the property that the cone over the continuum is homeomorphic to the hyperspace of subcontinua of the continuum. Among our results are the following theorems: (i)Such a finite-dimensional continuum must be atriodic and one-dimensional; (ii)if such a continuum is hereditarily decomposable, then it must be an arc, an arcwiseconnected circle-like continuum, or a compactification of either (0,1) or R with remainder either a circle or an arc. Examples are given to show that none of these possibilities may be omitted.

Mapping solenoids onto strongly self-entwined, circle-like continua

Mapping solenoids onto strongly self-entwined, circle-like continua

Continua which cannot be mapped onto any nonplaner circle-like continuum

Continua which cannot be mapped onto any nonplaner circle-like continuum

Concerning Non-Planar Circle-Like Continua

In this paper it is proved that if a circle-like continuum M cannot be embedded in the plane, then M is not a continuous image of any plane continuum (Theorem 5).Suppose that (S, ρ) is a metric space. A finite sequence of domains L1, L2, … , Ln is called a linear chain provided Li intersects Lj if and only if |i — j| ⩽ 1. If, in addition, there is a positive number ∊ such that, for each i, the diameter of Li is less than ∊, then the linear chain is called a linear ∊-chain. If for each positive number ∊ the continuum M can be covered by a linear ∊-chain, then M is said to be chainable (or snake-like) (2).