Finite-Support Calculus in Lowen Fuzzy Topology: Quotients, Support Strata, and Stability

We study the nonempty subsets of cardinality at most n of a Lowen fuzzy space, equipped with the final fuzzy topology of the support map. The singleton embedding and the all-in and hit operators connect this construction to the original fuzzy topology. Its strict levels lie between the finite Vietoris topology and the ordinary support-quotient topology; these bounds coincide at Hausdorff or Alexandrov levels, and at levels with Hausdorff Kolmogorov quotient. Two countable examples show separately that each bound can be strict. For topologically generated spaces, we identify the full final fuzzy topology with the lower-semicontinuous topology of the support quotient. Stabilized density and network-weight profiles are preserved at every level without a quotient assumption. In the Hausdorff generated setting, the map from the nth symmetric power to finite supports is a covering of degree n−1k−1 over the k-point stratum and has no natural section for n≥3. Exact fiber formulas and a collision cocycle describe the loss of support information under induced maps. Union and flattening yield graded operations that are nonexpansive for Hausdorff metrics; induced maps retain their Lipschitz bounds, and the construction is stable in Gromov–Hausdorff distance. We distinguish these results from compact-category compatibility with the classical finite-subset functor and exhibit nontopological fuzzy structures whose levelwise quotient bounds nevertheless coincide.