Increasing strong size properties and strong size block properties

Abstract

Let X be a continuum. The n-fold hyperspace C_n(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 C_n(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 an increasing strong size property provided that if μ is a strong size map for C_n(X) and t_0 ∈ [0, 1) is such that μ^−1(t_0) has property P, then so does μ^−1(t) for each t ∈ (t_0,1). 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.