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Continua with unique symmetric product

José G. AnayaEnrique Castañeda-AlvaradoAlejandro Illanes — 2013

Commentationes Mathematicae Universitatis Carolinae

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 connected...

On the hyperspace C n ( X ) / C n K ( X )

José G. AnayaEnrique Castañeda-AlvaradoJosé A. Martínez-Cortez — 2021

Commentationes Mathematicae Universitatis Carolinae

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 C n ( X ) : K 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.

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

José G. AnayaEnrique Castañeda-AlvaradoAlejandro Fuentes-Montes de OcaFernando Orozco-Zitli — 2018

Commentationes Mathematicae Universitatis Carolinae

A connected topological space Z is unicoherent provided that if Z = A B where A and B are closed connected subsets of Z , then A B is connected. Let Z be a unicoherent space, we say that z 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.

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