Positive topological entropy and $Δ$-weakly mixing sets

The notion of $\Delta$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $\Delta$-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of $\Delta$-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. Moreover, Our results generalize several well known results and also answer several open questions.