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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.
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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