On Jones' set function T and the property of Kelley for Hausdorff continua

We extend to Hausdorff continua the result of Camargo and Uzcátegui that says that if X is a metric continuum, Jones' set function T is continuous on singletons and T is idempotent on closed sets, then G={T({x})|x∈X} is a decomposition of X. We also present important implications of this result, a couple of them answer questions of the celebrated Houston Problem Book. We study Hausdorff continua with the property of Kelley and with the property of Kelley weakly. We establish a Hausdorff version of Jones' Aposyndetic Decomposition Theorem. To this end, we introduce the uniform property of Effros.