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The notion of $\Delta$-weakly mixing set is introduced, which shares similarproperties of weakly mixing sets. It is shown that if a dynamical system haspositive topological entropy, then the collection of $\Delta$-weakly mixingsets is residual in the closure of the collection of entropy sets in thehyperspace. The existence of $\Delta$-weakly mixing sets in a topologicaldynamical system admitting an ergodic invariant measure which is not measurabledistal is obtained. Moreover, Our results generalize several well known resultsand also answer several open questions.

Positive topological entropy and $\Delta$-weakly mixing sets