Disconnections of plane continua

Bajguz, W.

Displaying similar documents to “Disconnections of plane continua”

On topological factors of 3 dimensional locally connected continuum embeddable in $E^{3}$

Witold Rosicki (1978)

Fundamenta Mathematicae

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Continua with unique symmetric product

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Illanes (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a metric continuum. Let $F_{n} (X)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_{n} (X)$ provided that the following implication holds: if $Y$ is a continuum and $F_{n} (X)$ is homeomorphic to $F_{n} (Y)$ , then $X$ is homeomorphic to $Y$ . In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_{2} (X)$ , and (2) let $X$ be an arcwise...

A note on the paper ``Smoothness and the property of Kelley''

Gerardo Acosta, Álgebra Aguilar-Martínez (2007)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a continuum. In Proposition 31 of J.J. Charatonik and W.J. Charatonik, , Comment. Math. Univ. Carolin. (2000), no. 1, 123–132, it is claimed that $L (X) = ⋂_{p \in X} S (p)$ , where $L (X)$ is the set of points at which $X$ is locally connected and, for $p \in X$ , $a \in S (p)$ if and only if $X$ is smooth at $p$ with respect to $a$ . In this paper we show that such equality is incorrect and that the correct equality is $P (X) = ⋂_{p \in X} S (p)$ , where $P (X)$ is the set of points at which $X$ is connected im kleinen. We also use the correct equality to obtain some...

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...

Homeomorphisms of composants of Knaster continua

Sonja Štimac (2002)

Fundamenta Mathematicae

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The Knaster continuum $K_{p}$ is defined as the inverse limit of the pth degree tent map. On every composant of the Knaster continuum we introduce an order and we consider some special points of the composant. These are used to describe the structure of the composants. We then prove that, for any integer p ≥ 2, all composants of $K_{p}$ having no endpoints are homeomorphic. This generalizes Bandt’s result which concerns the case p = 2.

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.....................................

A differential equation related to the $l^{p}$ -norms

Jacek Bojarski, Tomasz Małolepszy, Janusz Matkowski (2011)

Annales Polonici Mathematici

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Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^{p}$ -distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

On the structure of homeomorphisms of the open annulus

Lucien Guillou (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $h$ be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that $h$ admits a $h$ -invariant dense open set $O$ on which it is conjugate to a translation and we study the action of $h$ on the compactly connected components of the closed and without interior set $R^{2} ∖ O$ .