  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.
