Attouch-Wets convergence for closed sets in $\star$-metric spaces via scalarization

We developed an Attouch-Wets convergence theory for nonempty closed subsets of $\star$-metric spaces, associated with an admissible scalarization and a bounded testing family. Since the nonlinear triangle structure of a $\star$-metric space does not permit a direct transfer of the classical bounded-Hausdorff framework, we first introduced admissible scalarizations that convert bounded-region distance-profile comparisons into a workable additive setting. On this basis, we defined profile and truncated excess functionals associated with a scalarization, established equivalent formulations of the resulting convergence, and showed that it is pseudometrizable whenever the testing family has a countable cofinal subfamily. We then compared this convergence with the corresponding bornological convergences and with Wijsman convergence, and showed that the latter is strictly weaker in general, while equivalence holds under properness of the linearized metric. We also proved restriction and product results, together with relative compactness and sequential completeness theorems for the induced hyperspace structure. The examples showed that the theory is nontrivial in genuinely nonlinear $\star$-metric settings and that the main assumptions used in the comparison and compactness results are essential.