Homeomorphisms of composants of Knaster continua

Sonja Štimac

Displaying similar documents to “Homeomorphisms of composants of Knaster continua”

Decompositions of the plane and the size of the continuum

Ramiro de la Vega (2009)

Fundamenta Mathematicae

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We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each $S_{i}$ intersects each $E_{i}$ -class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition...

Singular arc-like continua

Tadeusz Maćkowiak

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CONTENTSIntroduction.......................................................................................................51. Preliminaries.................................................................................................6 A. Mappings....................................................................................................6 B. Arc-like continua.........................................................................................8 C. Pseudosuspensions...................................................................................8 D....

On Dimensionsgrad, resolutions, and chainable continua

Michael G. Charalambous, Jerzy Krzempek (2010)

Fundamenta Mathematicae

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For each natural number n ≥ 1 and each pair of ordinals α,β with n ≤ α ≤ β ≤ ω(⁺), where ω(⁺) is the first ordinal of cardinality ⁺, we construct a continuum $S_{n, α, β}$ such that (a) $d i m S_{n, α, β} = n$ ; (b) $t r D g S_{n, α, β} = t r D g o S_{n, α, β} = α$ ; (c) $t r i n d S_{n, α, β} = t r I n d ₀ S_{n, α, β} = β$ ; (d) if β < ω(⁺), then $S_{n, α, β}$ is separable and first countable; (e) if n = 1, then $S_{n, α, β}$ can be made chainable or hereditarily decomposable; (f) if α = β < ω(⁺), then $S_{n, α, β}$ can be made hereditarily indecomposable; (g) if n = 1 and α = β < ω(⁺), then $S_{n, α, β}$ can be made chainable and hereditarily indecomposable. In...

Arcwise accessibility in hyperspaces

Sam B. Nadler, Jr.

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CONTENTS1. Introduction........................................................................................................ 52. Segmentwise accessibility..................................................................................... 73. Arcwise accessibility of singletons....................................................................... 84. Compacta in X which arcwise disconnect $2^{X}$ or C(X)................................ 155. Hereditary indecomposability and arcwise accessibility.....................................

Hereditarily indecomposable inverse limits of graphs

K. Kawamura, H. M. Tuncali, E. D. Tymchatyn (2005)

Fundamenta Mathematicae

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We prove the following theorem: Let G be a compact connected graph and let f: G → G be a piecewise linear surjection which satisfies the following condition: for each nondegenerate subcontinuum A of G, there is a positive integer n such that fⁿ(A) = G. Then, for each ε > 0, there is a map $f_{ε} : G \to G$ which is ε-close to f such that the inverse limit $(G, f_{ε})$ is hereditarily indecomposable.

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

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A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

Extending generalized Whitney maps

Ivan Lončar (2017)

Archivum Mathematicum

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For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Global continuum of positive solutions for discrete $p$ -Laplacian eigenvalue problems

Dingyong Bai, Yuming Chen (2015)

Applications of Mathematics

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We discuss the discrete $p$ -Laplacian eigenvalue problem, $\{\begin{matrix} Δ (φ_{p} (Δ u (k - 1))) + λ a (k) g (u (k)) = 0, k \in {1, 2, ..., T}, \\ u (0) = u (T + 1) = 0, \end{matrix}$ where $T > 1$ is a given positive integer and $φ_{p} (x) : = {| x |}^{p - 2} x$ , $p > 1$ . First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(λ, u) = (0, 0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $λ > 0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $λ > 0$ .

Making holes in the cone, suspension and hyperspaces of some continua

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Fuentes-Montes de Oca, Fernando Orozco-Zitli (2018)

Commentationes Mathematicae Universitatis Carolinae

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A connected topological space $Z$ is unicoherent provided that if $Z = A \cup B$ where $A$ and $B$ are closed connected subsets of $Z$ , then $A \cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z \in Z$ makes a hole in $Z$ if $Z - {z}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

On the hyperspace $C_{n} (X) / {C_{n}}_{K} (X)$

José G. Anaya, Enrique Castañeda-Alvarado, José A. Martínez-Cortez (2021)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a continuum and $n$ a positive integer. Let $C_{n} (X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$ , define the hyperspace ${C_{n}}_{K} (X) = {A \in C_{n} (X) : K \subset A}$ . In this paper, we consider the hyperspace $C_{K}^{n} (X) = C_{n} (X) / {C_{n}}_{K} (X)$ , which can be a tool to study the space $C_{n} (X)$ . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility. ...

Property of being semi-Kelley for the cartesian products and hyperspaces

Enrique Castañeda-Alvarado, Ivon Vidal-Escobar (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we construct a Kelley continuum $X$ such that $X \times [0, 1]$ is not semi-Kelley, this answers a question posed by J.J. Charatonik and W.J. Charatonik in A weaker form of the property of Kelley, Topology Proc. 23 (1998), 69–99. In addition, we show that the hyperspace $C (X)$ is not semi- Kelley. Further we show that small Whitney levels in $C (X)$ are not semi-Kelley, answering a question posed by A. Illanes in Problemas propuestos para el taller de Teoría de continuos y sus hiperespacios, Queretaro,...

CH and the Sacks property

S. Quickert (2002)

Fundamenta Mathematicae

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We show the consistency of CH and the statement “no ccc forcing has the Sacks property” and derive some consequences for ccc $ω^{ω}$ -bounding forcing notions.

Another ⋄-like principle

Michael Hrušák (2001)

Fundamenta Mathematicae

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A new ⋄-like principle $⋄_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg ⋄_{}$ is consistent with CH and that in many models of = ω₁ the principle $⋄_{}$ holds. As $⋄_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that $⋄_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁...