Increasing strong size properties and strong size block properties

Abstract

Let X be a continuum. The n-fold hyperspace Cn(X), n∈N, is the family of all nonempty closed subsets of X with at most n components, topologized with the Hausdorff metric. Let μ be a strong size map for Cn(X). A strong size level is the subset μ−1(t), with t∈[0,1]. A strong size block is the subset μ−1([s,t]), with 0≤s<r≤1. A topological property P is said to be a increasing strong size property provided that if μ is a strong size map for Cn(X) and t0∈[0,1), is such that μ−1(t0) has property P, then so does μ−1(t) for each t∈(t0,1). In this paper we show that uniform pathwise connectedness, uniform continuum-chainability and local connectedness are increasing strong size properties, and we will show some strong size block properties.