This paper concerns the behavior of F. B. Jones’ set function \(\mathcal{T}\), in connection with metrizable continua. For a continuum \(X\), let \(2^X\) denote its hyperspace of closed nonempty subsets, endowed with the Vietoris topology (induced by the Hausdorff metric), and let \(\mathcal{F}_1(X)\) be the subspace of singleton subsets. Given a subset \(A\) of \(X\), \(x\in X\) is in \(\mathcal{T} (A)\) precisely when each subcontinuum neighborhood of \(x\) intersects \(A\). \(\mathcal{T}(A)\) is always closed; so we may regard \(\mathcal{T}\) as mapping \(2^X\) to itself, and ask how the images \(\mathcal{T}(2^X))\) and \(\mathcal{T}(\mathcal{F}_1(X))\) behave as subspaces of \(2^X\). (For example, it is straightforward to show that \(|\mathcal{T}(\mathcal{F}_1(X))| = 1\) if and only if \(X\) is indecomposable.)
We quote from the authors’ summary: “…we are interested in when either \(\mathcal{T}(\mathcal{F}_1(X))\) or \(\mathcal{T} (2^X)\) is finite or countable. We introduce the notion of \(\omega\)-indecomposable continuum as a generalization of the well known concept of \(n\)-indecomposable continuum. We also present results about connectedness and compactness of \(\mathcal{T} (2^X)\). Finally, we give a generalization, to continua with the property of Kelley, of a couple of results known for homogeneous continua.”