Topological and Measure-Theoretic Properties of Partition-Dominating Sets (Preprint)

This paper extends partition-dominating sets—subsets whose elements lie in maximum-cardinality parts across partition families—to topological and measure-theoretic contexts. For countable partition families on Polish spaces with nowhere-dense rigging, we construct comeager dominating sets; for probability measure spaces with null rigging, we obtain conull dominating sets. For uncountable products indexed by κ, we introduce λ-scaled rigging (λ ≤ κ) ensuring complements of size at least λ⁺. Under measurable cardinals, the generated filter embeds in a measurable ultrafilter, yielding homogeneity properties. The results unify Baire category, measure, and large-cardinal frameworks, and all proofs are formalized in ZFC unless stated otherwise.",