The Hyperspace $$F_n^K(X)$$

Abstract

Given a metric continuum X and a positive integer n, $$F_{n}(X)$$ denotes the hyperspace of all nonempty subsets of X with at most n points endowed with the Hausdorff metric. For any $$K\in F_{n}(X)$$ , we define $$F_{n}(K,X)$$ as the collection of elements of $$F_{n}(X)$$ containing K and we consider $$F_{n}^K(X)$$ as the quotient space obtained from $$F_{n}(X)$$ by shrinking $$F_{n}(K,X)$$ 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 $$F_{n}^K(X)$$ .