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In this paper we construct a Kelley continuum such that 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 is not semi- Kelley. Further we show that small Whitney levels in 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, 2013.
Let be a metric continuum. Let denote the hyperspace of nonempty subsets of with at most elements. We say that the continuum has unique hyperspace provided that the following implication holds: if is a continuum and is homeomorphic to , then is homeomorphic to . In this paper we prove the following results: (1) if is an indecomposable continuum such that each nondegenerate proper subcontinuum of is an arc, then has unique hyperspace , and (2) let be an arcwise connected...
Let be a continuum and a positive integer. Let be the hyperspace of all nonempty closed subsets of with at most components, endowed with the Hausdorff metric. For compact subset of , define the hyperspace . In this paper, we consider the hyperspace , which can be a tool to study the space . 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.
Given a metric continuum and a positive integer , denotes the hyperspace of all nonempty subsets of with at most points endowed with the Hausdorff metric. For , denotes the set of elements of containing and denotes the quotient space obtained from by shrinking to one point set. Given a map between continua, denotes the induced map defined by . Let , we shall consider the induced map in the natural way . In this paper we consider the maps , , for some and for...
A connected topological space is unicoherent provided that if where and are closed connected subsets of , then is connected. Let be a unicoherent space, we say that makes a hole in if 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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