Hyperspaces through regular and meager subcontinua

Abstract

Given a metric continuum X, let C(X) and F1(X) be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let D(X) be the collection of all regular subsets in X belonging to C(X) and let M(X) be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: F1(X)=M(X), D(X)=D(X)−F1(X), D(X)=C(X)−M(X) and C(X)=D(X)∪M(X). We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite D1 are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function f:X→[0,1]. We show that the conditions that X is a λ-type continuum and every t in [0,1] is a cohesion point are equivalent to the equalities D(X)={A∈C(X):f(A)∈D([0,1])} and M(X)={A∈C(X):f(A)∈M([0,1])}. Throughout this paper we pose open problems.