Positive topological entropy and Δ-weakly mixing sets

The notion of Δ-weakly mixing set is introduced and studied. It is proved that Δ-weakly mixing sets share many properties with weakly mixing sets, in particular, if a dynamical system has positive topological entropy, then the collection of Δ-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of Δ-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. These results generalize several well known results and also answer several open questions.