Homogeneous continua for which the set function \(\mathcal T\) is continuous (Q860470)

\textit{F. Burton Jones} [Am. J. Math. 70, 403--413 (1948; Zbl 0035.10904)] defined the set function \({\mathcal T}\) and \textit{D. Bellamy} [Trans. Am. Math. Soc. 151, 581--587 (1970; Zbl 0207.53001)] gave conditions on a continuum for which \({\mathcal T}\) is continuous. The author of this paper in [Houston J. Math 32, No. 1, 161--165 (2006; Zbl 1102.54021)] gave a class of decomposable nonlocally connected one-dimensional continua for which \({\mathcal T}\) is continuous. In 2004 David Bellamy and Sam Nadler conjectured that if \(X\) is a homogeneous one-dimensional continuum, then \({\mathcal T}\) is continuous for \(X\). In this paper the author gives a negative answer to this question and establishes that \({\mathcal T}\) is continuous for a homogeneous continuum \(X\) if and only if one of the following conditions holds: 1. \(X\) is locally connected; 2. \(X\) is indecomposable; 3. \(X\) is not aposyndetic and is homeomorphic to the simple closed curve \(S^1\) or to the Menger universal curve \({\mathcal M}\), where \({\mathcal G}= \{{\mathcal T}(\{x\}): x\in X\}\).