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A Cantor set in the plane that is not σ-monotone

Aleš Nekvinda, Ondřej Zindulka (2011)

Fundamenta Mathematicae

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

A characterization of dendroids by the n-connectedness of the Whitney levels

Alejandro Illanes (1992)

Fundamenta Mathematicae

Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).

A class of continua that are not attractors of any IFS

Marcin Kulczycki, Magdalena Nowak (2012)

Open Mathematics

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

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