The hyperspace F_n^K(X)

Abstract

Given a metric continuum X and a positive integer n, Fn(X) denotes the hyperspace of all nonempty subsets of X with at most n points endowed with the Hausdorff metric. For any K ∈ Fn(X), we define Fn(K, X) as the collection of elements of Fn(X) containing K and we consider FKn (X) as the quotient space obtained from Fn(X) by shrinking Fn(K, X) to one point set, endowed with the quotient topology. In this paper, we report the first results of the investigation related with this hyperspace. We focus our attention on proving results regarding to aposyndesis, local connectedness, arcwise connectedness, unicoherence, and cut points of FKn(X).