About two definitions of F-sensitivity

Let (X, f) be a topological dynamical system and F be a Furstenberg family. There are two definitions about F-sensitivity of the system (X, f) but unfortunately, by Theorem 6 in Tan and Zhang (2011) [10], these are not equivalent in general. In this paper, we call these two definitions as F-orbit sensitivity and F-diameter sensitivity respectively. And then under the assumption that the dual family kF is countably generated we suggest sufficient and necessary conditions for F-diameter sensitivity and using this result, show that these two definitions are equivalent. (C) 2019 Elsevier B.V. All rights reserved.