Images of $\mathcal {T}$

Abstract

For a metric continuum X, we study possible images of the set function $\mathcal {T}$. We begin by showing that $\mathcal {T}(2^X)$ is an analytic set for every metric continuum X. We are interested when either $\mathcal {T}(\mathcal {F}_1(X))$ or $\mathcal {T}(2^X)$ is finite or countable. The notion of ω-indecomposable continuum is given as a generalization of the well known concept of n-indecomposable continuum. We also present results about the connectivity and compactness of $\mathcal {T}(2^X)$. We show that if X is a metric continuum such that $\mathcal {T}(2^X)$ is compact, then $\mathcal {T}(2^X)$ is either finite or uncountable. We give an example of a continuum X such that $\mathcal {T}(2^X)$ is countable.