Increasing strong size properties and strong size block properties

摘要

Let X be a continuum. The n-fold hyperspace Cn(X), n is an element of N, is the family of all nonempty closed subsets of X with at most n components, topologized with the Hausdorff metric. Let mu be a strong size map for Cn(X). A strong size level is the subset mu(-1)(t), with t is an element of [0, 1]. A strong size block is the subset mu(-1)([s,t]), with 0 s r 1. A topological property P is said to be a increasing strong size property provided that if mu is a strong size map for Cn(X) and t(0) is an element of [0, 1), is such that mu-1(t(0)) has property P, then so does mu-1(t) for each t is an element of (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. (C) 2020 Elsevier B.V. All rights reserved.